^STos^iiS^' 


THE 


ELEMENTS  OF  ALTERNATING  CURRENTS 


THE 


ELEMENTS 


OF 


ALTERNATING   CURRENTS 


BY 

W.   S.   FRANKLIN 

AND 

R.   B.  WILLIAMSON 


•       THE  MACMILLAN   COMPANY 

LONDON:  MACMILLAN  &  CO.,  Ltd. 
1899 

All  rights  reserved. 


Copyriglit,  1899 
By  W.   S.   franklin 


60113 


The  New  Era  Printing  Company 

Lancaster,  Pa. 

U.  8   A. 


PREFACE. 


This  book  represents  the  experience  of  seven  years'  teaching 
of  alternating  currents,  and  almost  every  chapter  has  been  sub- 
jected repeatedly  to  the  test  of  class-room  use.  The  authors 
have  endeavored  to  include  in  the  text  only  those  things 
which  contribute  to  the  fundamental  understanding  of  the 
subject  and  those  things  which  are  of  importance  in  the 
engineering   practice   of  to-day. 

It  may  be  taken  for  granted  that  the  authors  are  deeply  in- 
debted to  Mr.  C.  P.  Steinmetz,  whose  papers  are  unique  in  their 
close  touch  with  engineering  realities.  W.  S.  F. 

South  Bethlehem, 
June,  1899. 


TABLE    OF    CONTENTS. 


CHAPTER  I. 

Page. 
Magnetic  flux.     Induced  electromotive  force.     Inductance.     Capacity I 

CHAPTER  II. 

The  simple  alternator.     Alternating  e.  m.  f.  and  current.     The  contact  maker....     i8 

CHAPTER  III. 
Measurements  in  alternating  currents.     Ammeters.     Voltmeters.     Wattmeters....     29 

CHAPTER  IV. 
Harmonic  electromotive  force  and  current gg 

CHAPTER  V. 
Problem  of  the  inductive  circuit.     Problem  of  the  inductive  circuit  containing  a 
condenser.     Electrical  Resonance 51 

CHAPTER  VI. 
The  use  of  complex  quantity 62 

CHAPTER  VII. 
The  problem  of  coils  in  series.     The  problem  of  coils  in  parallel.     The  problem 
of  the  transformer  without  iron 69 

CHAPTER  VIII. 
Polyphase  alternators.     Polyphase  systems 80 

CHAPTER  IX. 
The  theory  of  the  alternator.     Alternator  designing 95 

CHAPTER  X. 
The  theory  of  the  transformer 119 

CHAPTER  XL 
Transformer  losses  and  efficiency.     Transformer  connections.     Transformer  de- 
signing   137 

CHAPTER  XII. 
The  synchronous  motor 151 

CHAPTER  XIII. 
The  rotary  converter 166 

CHAPTER  XIV. 
The  induction  motor 1 78 

CHAPTER  XV. 
Transmission  lines ,..., igS 


SYMBOLS. 


i  instantaneous  value  of  current, 
/  maximum  value  of  an  harmonic  alternating  current. 
/  effective  value  of  an  alternating  current. 
e  instantaneous  value  of  e.  m.  f 

M  maximum  value  of  an  harmonic  alternating  e.  m.  f. 
E  effective  value  of  an  alternating  e.  m.  f 
r   R  resistance  {r  sometimes  used  for  radius). 
L  inductance. 
J  electrostatic  capacity. 
t  time. 

T  Z  turns  of  wire. 

z  turns  of  wire  per  unit  length  of  a  coil. 
n  speed  in  revolutions  per  second. 
f  frequency  in  cycles  per  second. 
oi  frequency  in  radians  per  second. 
II  magnetic  permeability. 
/  length. 
q  sectional  area. 
N  magnetic  flux. 
B  flux  density. 


THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 


CHAPTER   I. 

INDUCTANCE  AND    CAPACITY. 

1.  Magnetic  flux. — Let  a  be  an  area  at  right  angles  to  the 
velocity  of  a  moving  fluid,  and  let  v  be  the  velocity  of  the  fluid. 
Then  av  is  the  flux  of  fluid  across  the  area  in  units  volume  per 
second.  Similarly  the  product  of  the  intensity,  /,  of  a  magnetic 
field  into  an  area  a  at  right  angles  to /is  called  the  magnetic  flux 
across  the  area.     That  is 

N=fa  (i) 

in  which  N  is  the  magnetic  flux  across  an  area  a  which  is  at  right 
angles  to  a  magnetic  field  of  intensity/. 

The  unit  of  magnetic  flux  is  the  flux  across  one  square  centi- 
meter of  area  at  right  angles  to  a  magnetic  field  of  unit  intensity. 
This  unit  flux  is  called  a  line  of  force  *  or  simply  a  line.  For 
example,  the  intensity  of  the  magnetic  field  in  the  air  gap  between 
the  pole  face  of  a  dynamo  and  the  armature  core  is,  say,  5000 
units,  and  this  field  is  normal  to  the  pole  face  of  which  the  area 
is  300  square  centimeters,  so  that  1,500,000  lines  of  magnetic 
flux  pass  from  the  pole  face  into  the  armature  core. 

The  trend  of  the  lines  of  force  near  the  poles  of  a  magnet  is 
shown  in  Fig.  i.  In  Fig.  2  is  shown  the  trend  of  the  lines  of 
force  through  a  coil  of  wire  in  which  an  electric  current  is  flowing. 

*  A  line  of  force  is  a  line  drawn  in  a  magnetic  field  so  as  to  be  in  the  direction  of 
the  field  at  each  point.  The  term  line  of  force  is  used  for  the  unit  flux  for  the  follow- 
ing reason :  Consider  a  magnetic  field.  Imagine  a  surface  drawn  across  this  field. 
Suppose  this  surface  to  be  divided  into  parts  across  each  of  which  there  is  unit  flux. 
Imagine  lines  of  force  drawn  in  the  magnetic  field  so  that  one  line  of  force  passes 
through  each  of  the  parts  of  our  surface.  Then  the  magnetic  flux  across  any  area 
anywhere  in  th&  field  will  be  equal  to  the  miinher  of  these  lines  which  cross  the  area. 


THE   ELEMENTS   OF   ALTERNATIiSfG   CURRENTS. 


Magnetic  flux  through  a  coil. — In  the  discussion  of  the  induc- 
tances of  coils,  it  is  customary  to  speak  of  the  magnetic  flux 
through  a  coil  as  the  product  of  the  number  of  hues  through  the 


Fig.  I. 

opening  *  of  the  coil  into  the  number  of  turns  of  wire  in  the  coil ; 

that  is,  the  lines  are  counted  as  many  times  as  there  are  turns  of 

wire. 

2.  Induced  electromotive  force. — - 
When  a  bundle  of  Z  wires  con- 
nected in  series  moves  across  a 
magnetic  field  so  as  to  cut  the  lines 
of  force,  in  each  wire  an  e.  m.  f.  is 
induced  which  is  equal  to  the  rate, 
dN 


,  at  which  lines  of  force  are  cut, 


dt 

and  the  total  e.  m.  f.  induced  in  the 
bundle  of  wires  is 

dN 


e=-Z 


dt 


(") 


Similarly  when  the  magnetic  flux  through  the  opening  of  a  coil 
changes  an  e.  m.  f  is  induced  in  the  coil,  such  that 


^dN 
'=-^-dt 


(ii)  bis 


dN 


in  which  Z  is  the  number  of  turns  of  wire  in  the  coil,  -^   is  the 


*  Strictly,  the  number  of  lines  through  a  mean  turn  multiplied  by  the  number  of 
turns. 


INDUCTANCE   AND    CAPACITY.  3 

rate  of  change  of  the  flux,  and  e  is  the  induced  e.  m.  f  The 
negative  sign  is  chosen  for  the  reason  that  an  increasing  positive 
flux  produces  a  left-handed  e.  m.  i.  in  the  coil.* 

Examples. — {a)  A  conductor  on  a  dynamo  armature  cuts  the 
1,500,000  lines  of  force  from  one  pole  face  in,  say,  -Jq-  second, 
that  is,  at  the  rate  of  75,000,000  lines  per  second  ;  and  this  is  the 
e.  m.  f  (in  c.  g.  s.  units)  induced  in  the  conductor. 

{p)  A  coil   Z,   Fig.    I,    having  Z  turns  of  wire  surrounds   a 

magnet  NS  through  which  there  are  N  lines  of  flux.     The  coil 

is  quickly  removed  from  the  magnet,  reversed,  and  replaced ;  the 

whole  operation  being  accomplished  in  t  seconds.     The  flux  tV 

being  reversed  with  respect  to  the  coil  is  to  be  considered  as 

changing   -\-  N  to  —  N,  the  total  change  being  therefore  2N. 

2N 
Dividing  this  total  change  of  flux  by  the  time  t  gives  — ,  which 

is  the  average  value  of  —r    dunng  the  time  /,  and  the  average 

.    .     2N 
e.  m.  f  induced  in  the  coil  is    — .  Z.     This  e.  m.  f  is  expressed 

in  c.  g.  s.  units  and  is  to  be  divided  by  10^  to  reduce  it  to  volts. 

3.  The   magnetic   field   as    a    seat    of    kinetic    energy. — The 

magnetic  field  is  a  kind  of  obscure  motion  of  the  all  pervading 
medium,  the  ether ;  and  this  motion  represents  energy.  The 
amount  of  energy  in  a  given  portion  of  a  magnetic  field  is  pro- 
portional to  the  square  of  the  intensity  of  the  field.  This  is 
analogous  to  the  fact  that  the  kinetic  energy  of  a  portion  of  a 
moving  Hquid  is  proportional  to  the  square  of  the  velocity  of  the 
liquid. 

4.  Kinetic  energy  of  the  electric  current  in  a  coil.  Defini- 
tion of  Inductance. — The  kinetic  energy  of  an  electric  current  is 
the  energy  which  resides  in  the  magnetic  field  produced  by  the 
current.  The  kinetic  energy  is,  at  each  point,  proportional  to  the 
square  of  the  field  intensity,  that  is,  to  the  square  of  the  current. 

*  This,  although  an  inadequate  statement,  must  suffice  ;  especially  inasmuch  as  the 
sign  in  equation  (ii)  is  of  no  practical  importance. 


4  THE   ELEMENTS   OF   ALTERNATING   CURRENTS,. 

Therefore  the  total  kinetic  energy  of  the  field  is  proportional  to 
the  square  of  the  current.     That  is, 

JV=  %LP  (i) 

in  which  W  is  the  total  energy  of  a  current  z  in  a  given  coil,  and 
(^Z)  is  the  proportionality  factor.  The  quantity  L  is  called  the 
inductance  of  the  given  coil. 

Units  of  'Inductance. — When  in  equation  (i)  Wis,  expressed 
in  joules  and  i  in  amperes,  then  L  is  expressed  in  terms  of  a 
unit  called  the  henry.  When  W  is  expressed  in  ergs  and  i  in 
c.  g,  s.  units  of  current,  then  L  is  expressed  in  c.  g.  s.  units  of  in- 
ductance. The  c.  g.  s.  unit  of  inductance  is  called  the  centimeter, 
for  the  reason  that  the  square  of  a  current  must  be  multiplied 
by  a  length  to  give  energy  or  work  ;  that  is,  inductance  is  ex- 
pressed as  a  length  and  the  unit  of  inductance  is,  of  course,  the 
unit  of  length.  The  henry  is  equal  to  lo^  centimeters  of  induc- 
tance. 

Example :  A  given  coil  with  a  current  of  0.8  c.  g.  s.  units 
produces  a  magnetic  field  of  which  the  total  energy  is  6,400,000 
ergs,  so  that  the  value  of  L  for  this  coil  is  20,000,000  centime- 
ters. If  the  current  is  expressed  in  amperes  and  energy  in  joules 
then  the  total  energy  corresponding  to  8  amperes  would  be  0.64 
joules  and  the  value  of  L  would  be  0.02  henry. 

Non-inductive  circuits:  A  circuit  of  which  the  inductance  is 
negligibly  small  is  called  a  non-inductive  circidt.  Since  the  in- 
ductance of  a  circuit  depends  upon  the  energy  of  the  magnetic 


lamys-^t^  (^  (^  (^ 


Fig.  3.       . 

field,  therefore  a  non-inductive  circuit  is  one  which  produces  only 
a  weak  field,  or  a  field  which  is  confined  to  a  very  small  region. 
Thus,  the  two  wires.  Fig.  3,  constitute  a  non-inductive  circuit, 
especially  if  they  are  near  together ;  for  these  two  wires  with  op- 


INDUCTANCE   AND    CAPACITY.  5 

posite  currents  produce  only  a  very  feeble  magnetic  field  in  the 
surrounding  region.  The  wires  used  in  resistance  boxes  are 
usually  arranged  non-inductively.  This  may  be  done  by  doub- 
ling the  wire  back  on  itself  and  winding  this  double  wire  on  a 
spool.  In  this  case  the  e.  m.  f  between  adjacent  wires  may  be 
great  and  they  may  have  considerable  electrostatic  capacity.  In 
order  to  make  a  non-inductive  resistance  coil  without  this  de- 
fect, the  wire  may  be  wound,  in  one  layer,  on  a  thin  paper  cylin- 
der so  as  to  bring  the  terminals  as  far  apart  as  possible.  This 
cylindrical  coil  is  then  flattened  so  as  to  reduce  the  region  (in- 
side) in  which  the  magnetic  field  is  intense.  This  gives  a  non- 
inductive  coil  of  which  the  electrostatic  capacity  is  inconsiderable. 

5.  Moment  of  inertia,  analogue  of  inductance. — The  kinetic  en- 
ergy of  a  rotating  wheel  resides  in  the  various  moving  particles 
of  the  wheel.  The  velocity  (Hnear)  of  each  particle  of  the  wheel 
is  proportional  to  the  speed  (angular  velocity)  of  the  wheel,  and 
the  energy  of  each  particle  is  proportional  to  the  square  of  its 
velocity,  that  is,  to  the  square  of  the  speed.  Therefore,  the  total 
kinetic  energy  of  the  wheel  is  proportional  to  the  square  of  the 
speed.     That  is, 

IV=  %Koi''  (2) 

in  which  W  is  the  total  energy  of  a  wheel  rotating  at  angular  ve- 
locity CO  and  (^^)  is  the  proportionality  factor.  The  quantity 
K\s  called  the  moment  of  inertia  of  the  wheel. 

6.  Proposition. — The  inductance  of  a  coil  wound  07t  a  given  spool 
is  proportional  to  the  square  of  the  number  of  turns,  Z,  of  wire. 
For  example,  a  given  spool  wound  with  No.  16  wire  has  500 
turns  and  an  inductance  of  say,  0.025  henry;  the  same  spool 
wound  with  No.  28  wire  would  have  about  ten  times  as  many 
turns  and  its  inductance  would  be  about  100  times  as  great  or 
2.5  henrys. 

Proof :  To  double  the  number  of  turns  on  a  given  spool  would  everywhere  double 
the  field  intensity  for  the  same  current,  and  therefore  the  energy  of  the  field  would 
everywhere  be  quadrupled  for  a  given  current  so  that  the  inductance  would  be  quad- 
rupled according  to  equation  (i). 


6     THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

7.  Proposition. — TJie  inductance  of  a  coil  of  given  shape  is  pro- 
portional to  its  linear  dimensions,  the  number  of  turns  of  wire  be- 
ing unchanged.  For  example,  a  given  coil  has  an  inductance  of 
0.022  henry,  and  a  coil  three  times  as  large  in  length,  diameter, 
etc.,  has  an  inductance  of  0.066  henry. 

8.  Electro-motive  force  required  to  make  a  current  in  a  coil 
change. — A  current  once  established  in  a  coil  of  zero  resistance 
would  continue  to  flow  without  the  help  of  an  e.  m.  f.  to  main- 
tain it  just  as  a  wheel  when  once  started  continues  to  turn,  pro- 
vided there  is  no  resistance  to  the  motion  of  the  wheel.  To  in- 
crease the  speed  of  the  wheel  a  torque  must  act  upon  it  in  the 
direction  of  its  rotation,  and  to  increase  the  current  in  the  coil 
an  e.  m.  f.  must  act  on  the  coil  in  the  direction  of  the  current. 

When  an  e.  m.  f.  e  (over  and  above  the  e.  m.  f.  required  to 

overcome  the  resistance  of  the  coil)  acts  upon  a  coil  the  current 

di 
is  made  to  increase  at  a  definite  rate,  -,  ,  such  that 

di 
'-^dt  (3)a 

Proof  of  equation  (3)  :  Multiplying  both  members  of  this  equation  by  the  current 

di  dW 

i  we  have  ei  ^=  Li  ~  .     Now  ei  is  the  rate,  — ;-  ,  at  which  work  is  done   on  the  coil 
dt  '    dt  ' 

in  addition  to  the  work  used  to  overcome  resistance,  and  this  must  be  equal  to  the 

rate  at  which  the  kinetic  energy  of  the  current  in  the  coil  increases.     Differentiating 

dW  di 

equation   (l)  we  have  — —  =  Zz  —  .     Therefore,  equation  (3)  is  proven. 

Torque  reqidred  to  made  the  speed  of  a  wheel  increase  :  When 
a  torque,  T  (over  and  above  the  torque  required  to  overcome 
the  frictional  resistance),  acts  upon  a  wheel,  then  the  angular  ve- 

dca 

locity,  CO,  of  the  wheel  is  made  to  increase  at  a  definite  rate  —r 

such  that 


Proof  of  equation  (4)  :  Multiplying  both  members  of  this  equation  by  the  angular 

du      ^^        ^     .     n  dW 

—  0     Now  7w  IS  the  rate  —  ,- 
dt  dt 


velocity,  u,  of  the  wheel  we  have  7w  ^Kcd  -—  =     Now  Tu  is  the  rate  ->-  at  which 


INDUCTANCE   AND   CAPACITY.  7 

work  is  done  on  the  wheel  and  this  must  be  equal  to  the  rate  at  which  the  kinetic  en- 

dW  ,^  du> 
—z-^Au-r. 
dt  dt 


^    ,         ,         .  ^.^^         .    .  ■        ,    \  1         dW       ^^  du> 

ergy  of  the  wheel  increases.     Dmerentiating  equation  (2j  we  nave  — -^Aw— -, 


Therefore  equation  (4)  is  proven. 

9.  Mag'netic  flux  through,  a  coil  due  to  a  current  in  the  coil. — In 

dealing  with  coils  it  is  usual  to  speak  of  the  magnetic  flux  tlu'OMgli 
the  coil  as  the  product  of  the  flux  through  the  opening  of  the 
coil  or  the  flux  through  a  mean  turn,  multiplied  by  the  number 
of  turns  as  pointed  out  in  Art.  i.     That  is 

N=ZN' 

in  which  N'  is  the  flux  through  the  opening  of  a  coil  (through 
a  mean  turn),  Z  is  the  number  of  turns  of  wire  in  the  coil  and 
iVis  what  is  called  \\^&  flux  through  the  coil. 

Proposition. — The  flux  N  through  a  coil  due  to  a  current  i  in 
in  the  coil  is 

N=Li  (5)* 

in  which  L  is  the  inductance  of  the  coil.  This  proposition  is 
proven  in  the  next  article. 

10.  Self-induced  e.  m.  f.  Reaction  of  a  changing  current. — When 
one  pushes  on  a  wheel,  causing  its  speed  to  increase,  the  wheel 
reacts  and  pushes  back  against  the  hand.     This  reacting  torque 

is  equal  and  opposite  to  the  acting  torque  K-r-  [equation  (4)] , 

which  is  causing  the  increase  of  speed.  Thus,  when  the  speed  of 
the  wheel  is  increasing,  the  reacting  torque  is  in  a  direction  op- 
posite to  the  speed,  and,  when  the  speed  is  decreasing,  the  reacting 
torque  is  in  the  same  direction  as  the  speed. 

Similarly  when  an  e.  m.  f.  acts  upon  a  circuit, f  causing  the 
current  to  increase,  the  increasing  current  reacts.     The  reacting 

di 
e.  m.  f.  is  equal  and  opposite  to  the  actmg  e.  m.  f.,  L--j  [equation 

(3)],  which  is  causing  the  current  to  increase.     This  reacting  e. 

*  In  this  equation  L  and  i  must  be  expressed  in  c.  g.  s.  units  because  the  unit  of 
flux  corresponding  to  the  ampere-henry  is  not  mitch  used. 

■f  Supposed  to  have  zero  resistance  for  the  sake  of  simplicity  of.  statement. 


8  THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

m.  f.  is  called  a  self-induced  e.  m.  f.  The  self-induced  e.  m.  f.  is 
therefore 

When  a  current  is  increasing  I  -j  positive  i  the  self-induced 
e.  m.  f.  is  opposed  to  the  current,  and  when  a  current  is  decreas- 
ing I  -J  negative  J  the  self-induced  e.  m.  f.  is  in  the  direction  of 
the  current,  exactly  as  in  the  case  of  a  rotating  wheel. 

Proof  of  equation  (5):    If  the  current  i  is  changing  then  from  equation  (S)  we 

have  — ;-  =  L — ,  but r-  is  an  e.  m.  f.   e  induced  in  the  coil  by  the  changing  flux 

dt  dt'  di  :  &    & 

and,  therefore,  by  the  changing  current.    That  is,  e  =  —  L—  ,  which,  being  identical 

to  equation  (3)b,  shows  that  equation  (5)  is  true. 

11.  Calculation  of  inductance  in  terms  of  magnetic  flux  per  unit 

current. — According  to  equation  (5)  the  inductance  of'  a  coil  is 

N 
equal  to  the  quotient  —  when  N  is  the  magnetic  flux  through 

the  coil  *  due  to  the  current  i  in  the  coil.  There  are  important 
cases  in  which  the  flux  through  a  coil  due  to  a  given  current  may 
be  easily  calculated  and,  therefore,  the  inductance  of  such  a  coil  is 
easily  determined. 

Long  Solenoid. — Consider  a  long  cylindrical  coil  of  wire  of 
mean  radius  r,  of  length  /  and  having  z  turns  of  wire  per  unit 
length.  The  field  intensity  in  the  coil  is  /=  /pizi  and  the  area 
of  the  opening  of  the  coil  is  tit^  so  that  the  flux  through  the 
opening  is  4.7z^r'^zi(=  N').  The  coil  has  Iz  turns,  so  that  N  = 
Iz.N'  =  471 W//;  dividing  this  by  i  we  have,  according  to  equa-' 
tion  (5), 

L  =  ^Tt^rhH  (6) 

This  equation  is  strictly  true  only  for  very  long  coils   on  which 

the  wire  is  wound  in  a  thin  layer  ;  the  equation  is,  however,  very 

*  That  is,  the  flux  through  a  mean  turn  multiplied  by  the  number  of  turns  of  wire. 


INDUCTANCE   AND    CAPACITY. 


9 


useful  in  enabling  one  to  calculate  easily  the  approximate  induc- 
tance of  even  short  thick  coils. 

Coil  wound  on  an  iron  core. — A 
coil  of  ^  turns  of  wire  is  wound  on 
an  iron  ring  /  cm.  in  circumference 
(mean)  and  q  cva?  in  sectional  area, 
as  shown  in  Fig.  4.  The  coil  pro- 
duces through  the  ring  a  magnetic 

in.  in.f. 
flux  N'  = ,  where  in.  in.  f. 


in.  r. 


(=  4.7tZi)  is  the  magneto-motive 
force  due  to  the  coil,  and  in.  r. 
I 


Fig.  4. 


(=  —  -  —  i  is  the  magnetic  reluctance  of  the  iron  core,  i  being;  the 
ft     q)  '='  >  ^ 

current  in  the  coil  and  //  the  permeability  of  the  iron.     Therefore, 


N=  ZN'  = 


47T/JLqZ^Z 


=  Lt 


or 


Z  = 


4.7rfj.qZ^ 


(7) 


Remark :  The  permeability  /^  of  iron  decreases  with  increasing 
magnetizing  force.  Therefore,  the  inductance  of  a  coil  wound  on 
an  iron  core  is  not  a  definite  constant  as  in  case  of  a  coil  without 
an  iron  core. 

12.  Growth  and  decay  of  current  in  an  inductive  circuit. — 
When  a  torque  is  applied  to  a  wheel  the  wheel  gains  speed  until 
the  whole  of  the  applied  torque  is  used  to  overcome  the  resist- 
ance of  the  air,  etc.  While  the  speed  is  increasing  part  of  the 
applied  torque  overcomes  this  resistance  and  the  remainder  causes 
the  speed  to  increase. 

When  an  electromotive  force  is  applied  to  a  circuit  the  current 
in  the  circuit  increases  until  the  whole  of  the  applied  e.  m.  f  is 
used  to  overcome  the  resistance  of  the  circuit.  While  the  cur- 
rent is  growing  part  of  the  applied  e.  m.  f  overcomes  resistance 
and  the  remainder  causes  the  current  to  increase.     Therefore, 


lO         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

i?=i?.  +  i§  (8) 

in  which  E  is  the  apphed  e.  m.  f.,  i,  is  the  instantaneous  value  of 

the  growing  current,  R  is  the  resistance  of  the  circuit  and  L  its 

inductance.     Ri  is  the  part  of  E  used  to  overcome  resistance  and 

di  . 
Z  -7  is  the  part  of  E  used  to  make  the  current  increase. 
ai 

If  a  circuit  of  inductance  L  and  resistance  R  with  a  given  current 

is  left  to  itself  without  any  e.  m.  f.  to  maintain  the  current  the 

current  dies  away  or  decays,  and  the  e.  m.  f ,  Ri,  which,  at  each 

instant   overcomes    the    resistance,  is    the    self-induced    e.  m.  f. 

y.  di  ,  ,   .  „ .  ^di 

—  L-j  -.so  that  at  each  instant  Rt=  —  L-r   or 

dt  dt 

^.       -^  di  ,  . 

o  =  R,  +  Lj^  (9) 

Examples :  An  e.  m.  f  of  i  lo  volts  acts  on  a  coil  of  which  the 

inductance  is  0.04  henry  and  the  resistance  is  3  ohms.     At  the 

instant  that  the  e.  m.  f.  begins  to  act  the  actual  current  i  in  the 

coil  is  zero  and  the  whole  of  the  e.  m.  f.  acts  to  increase  the  cur- 

,  ,  .  di       di 

rent,  so  that  1 10  volts  =  0.24  henry  x  -7-  or  —  =  2750  amperes 

per  second.     When  the  growing  current  has  reached  a  value  of 

30  amperes  Ri  is  equal  to  90  volts  and  the  remainder  of  the  1 10 

acts  to  cause  the  current  to  increase,  that  is,   20  volts  =  0.04 

,  di       di  . 

henry  x  -7-  or  —  =  500  amperes  per  second. 

If  a  current  is  established  in  this  coil  and  the  coil  left  to  itself, 
short  circuited,  without  any  e.  m.  f.  to  maintain  the  current ; 
then,  as  the  decaying  current  reaches  a  value  of,  say,  30  am- 
peres, the  e.  m.  f.  Ri  is  90  volts  and  this  e.  m.  f.  is  equal  to 

^  di         .       di  .  . 

—  Z—  so  that  —  IS  —  22i;o  amperes  per  second. 

dt  dt  0  I'         V 

13.  Problem  I. — An  inductive  circuit  with  a  current  flowing  in 
it  is  left  to  itself,  short  circuited.     At  a  certain  instant,  from  which 


INDUCTANCE   AND    CAPACITY. 


II 


time  is  to  be  reckoned  {t  =  o),  the  value  of  the  current  is  I.  It 
is  required  to  find  an  expression  for  the  decaying  current  at  each 
succeeding  instant ;  the  resistance  R  and  the  inductance  L  of  the 
circuit  being  given. 

Let  i  be  the  value  of  the  current  at  the  instant  t.     Then 


i  =  le 


(lo) 


Proof:  To  establish  the  truth  of  equation  (lo)  it  is  sufficient  to  show  that  i^  1 
when  ^  =  o,  and  that  equation  (9)  is  satisfied.     Substituting /^  =  o  in  equation  (10) 


we   have    i=I.      Differentiating   equation   (10)    we   have 


di 
It 


R     - 

Tie 


di_ 
di 


—  -i  ox  Ri-V  L  —^  =0  which  is  equation  (9). 


The  ordinates  of  the  curve  Fig.  5,  show  a  decaying  current. 


DECAYING    CURRENT 
I  =S6.7amp. 

L  =  O.04henrj/ 


handredili^  cf  a  Second 

Fig.  5. 


14.  Problem  II.  — A  constant  e.  m.  f.  E  is  connected  to  a  cir- 
cuit of  resistance  R  and  inductance  L.  Required  an  expression 
for  the  growing  current  /  seconds  after  the  e,  m.  f.  is  connected 
to  the  circuit. 

The  required  expression  is 


12 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


F)-oof:  To  establish  the  truth  of  equation  ( 1 1 )  it  is  sufficient  to  show  that  ?'  =  o 
when  /  =  o  and  that  equation  (8)  is  satisfied. 


GROWING  CURRENT 
E  =  110  voli^ 
R  =  3  ohmp 
L  =  OM  Mnry 

iime 


Jiu.ndredih$  of  a  Second 


Fig.  6. 


The  ordinates  of  the  curve,  Fig.  6,  show  the  values  of  a  grow- 
ing current. 

15.  Energy  of  two  coils.  Mu- 
tual Inductance.  Consider  two  sep- 
arate coils  in  which  currents  i'  and  i" 
respectively  are  flowing.  The  magnetic 
field  f",  Fig.  7,  at  any  given  point  p 
in  the  neighborhood  is  the  resultant  of 
the  field  intensities  f  sxi^f"  due  to 
the  respective  currents  so  that  from 
Pj(j^  7_  trigonometry 

f//2=fnj^  fn  _j_  2ff  cos  0 

Now  the  kinetic  energy  at  p  is  proportional  to  /'"''■  or  Xa  f^  -\-f'^  +  "i-ff"  cos  ^  so 
that  the  energy  at  the  point  consists  of  three  parts  proportional  respectively  to/'2,  to 
f''-  and  \.o  ff".  But  the  field  intensities  are  proportional  to  the  currents  in  the  two 
coils  so  that  the  three  parts  of  the  energy  at/  are  proportional  respectively  to  i'^  to 
i"'^  and  to  i'  i".  The  same  is  true  of  the  energy  at  every  other  point  of  the  magnetic 
field  so  that  the  total  kinetic  energy  of  two  coils  is  in  three  parts  which  are  propor- 
tional respectively  to  i'"^,  to  i""^  and  to  i'  i",  that  is, 

fF=  >^  Z^2'2  +  >^  U'i"^  +  MiH"  (12) 

in  which  WSs,  the  total  kinetic  energy  of  the  two  coils,  i'  and  i^'  the  currents  in  the 
coils  and  Y^L' ,  yiU'  and  il/are  proportionality  factors.  The  factors  U  and  L"  are 
the  self  inductances  of  the  respective  coils  (previously  defined)  and  the  factor  M  is 
called  the  mutual  inductance  of  the  two  coils. 


INDUCTANCE   AND    CAPACITY.  1 3 

From  equation  (12)  it  can  be  shown  that 

di'  \ 
dt  I 

(13) 
di^^  ( 
o?  e^^=  —  M-—  ] 

dt  j 

di^ 
in  which  e^'  is  the  e.  m.  f.  induced  in  one  coil  by  a  changing  current  —  in  the  other 

coil. 

From  equation  (13)  it  can  be  shown  that 

\  (14) 

and  N'  =  Mi") 

in  which  N^  is  the  flux  through  one  coil  (lines  of  force  counted  as  many  times  as 
there  are  turns  of  wire)  due  to  a  current  i'  in  the  other  coil. 

Examples  :  The  induction  coil  and  the  transformer  depend  for  their  action  upon 
the  mutual  inductance  of  two  coils,  the  primary  and  the  secondary  coils.  In  the 
theory  and  design  of  transformers  it  is  not  necessary  to  make  explicit  use  of  the  idea 
of  mutual  inductance. 

16.  Electric  charge. — The  electric  current  in  a  wire  is  looked 
upon  as  a  transfer  of  electric  charge  along  the  wire.  The  amount 
of  electric  charge  Q  which  in  t  seconds  passes  a  given  point  of  a 
wire  carrying  a  current  i  is 

Q  =  it  (15) 

or  the  rate  -^  at  which  the  charge  passes  a  given  point  on  a 

wire  is 

dQ       .  ,  .. 

^  =  ^-  ^^^) 

Units  charge. — When  i  ia  equation  (15)  is  expressed  in  am- 
peres and  t  in  seconds,  Q  is  expressed  in  terms  of  a  unit  called 
the  coulomb.  That  is,  the  coulomb  is  the  amount  of  electric 
charge  which  passes  in  one  second  along  a  wire  carrying  one  am- 
pere. When  i  is  expressed  in  c,  g.  s.  units  and  t  in  seconds,  Q 
is  expressed  in  terms  of  the  c.  g.  s.  unit  charge. 

Measurement  of  electric  charge. — An  electric  charge  may  be 
determined  by  measuring  the  current  i  which  it  will  maintain  dur- 
ing an  observed  time  t.  Then  Q  may  be  calculated  from  equa- 
tion  (15).     The  charge   capacity  of  storage   batteries   is   deter- 


14       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

mined  in  this  way.  A  very  small  charge  cannot  be  measured 
by  measuring  the  current  i  and  the  time  t,  for  such  a  charge  can- 
not maintain  a  steady  measurable  current  for  a  sufficient  time. 
A  small  electric  charge  is  measured  by  allowing  it  to  pass  quickly 
through  a  galvanometer  and  observing  the  throw  of  the  needle. 
The  charge  is  sensibly  proportional  to  the  throw.  A  galvanom- 
eter used  in  this  way  is  called  a  balhstic  galvanometer. 

17.  Condensers.  Electrostatic  capacity. — When  the  terminals 
of  a  battery  are  connected  to  two  metal  plates,  as  shown  in  Fig. 
8,  a  momentary  current  flows  as  indicated  by  the  arrows  and  the 

electric  charge  which  passes  along  the 
wire  during  this  momentary  current  is 
stored  upon  the  plates,  for  upon  dis- 
connecting the  battery  and  connecting 
the  plates  with  a  wire  a  momentary 
reversed  current  may  be  observed. 
If  a  ballistic  galvanometer  be  included 
in  the  circuit  the  amount  of  charge 
FiG_  g_  which  passes  into  the  plates  may  be 

measured.  This  amount  of  charge  is 
proportional  to  the  e.  m.  f  e,  of  the  battery  (other  things  being 
equal)  that  is 

Q=Je  (17) 

in  which  Q  is  the  electric  charge  which  flows  along  the  wire 
into  the  plates,  e  is  the  e.  m.  f  of  the  battery  and  /  the  propor- 
tionality factor.  Two  plates  arranged  in  this  way  constitute  what 
is  called  a  condenser  and  the  factor  /  is  called  the  electrostatic  ca- 
pacity of  the  condenser.  If,  in  equation  (17)  0  is  expressed  in 
coulumbs  and  e  in  volts,  then  /  is  expressed  in  terms  of  a  unit 
called  2.  farad.  That  is,  a  condenser  has  a  capacity  of  one  farad 
when  one  coulomb  of  electric  charge  is  pushed  into  it  by  a  bat- 
tery of  which  the  e.  m.  f  is  one  volt.  The  unit  of  capacity 
which  is  commonly  used  to  express  the  capacities  of  condensers, 
electric  cables,  etc.,  is  the  microfarad.     The  microfarad  is  one 


INDUCTANCE   AND    CAPACITY. 


15 


Fig.  9. 


millionth  of  a  farad.     The  microfarad  is  used  because  the  farad 
is  too  large  a  unit  to  use  conveniently. 

Condensers  to  have  a  large 
capacity  (as  much  as  a  micro- 
farad) are  usually  made  up  of 
alternate  sheets  of  tinfoil  and 
waxed  paper  or  mica,  as  indicated 

in   Fig.   9.     Alternate  metal    sheets  are    connected    together  as 
shown,  thus  practically  forming  two  plates  of  large  area. 

If  capacities  y^  _/2' Ts'  ^^^■>  ^^^  connected  in  para//e/  their  com- 
bined capacity  yis  equal  to  Ji-\-/2-h/s+^^c-  If  capacities  J^J^J^, 
etc.,  are  connected  in  series  their  combined  capacity  y  is  obtained 

I 

-I-  etc. 


It  will  be  noted 


I        I         I 
from  the  expression  —  =  y  -|-  y  4-  y 

that  capacities  combine  in  a  manner  just  the  opposite  of  that  of 
resistances. 

18.  Mechanical  and  electrical  analogies. — The  analogy  between  moment 
of  inertia  and  inductance  as  pointed  out  in  t"he  discussion  of  inductance  is  but  a  small 
part  of  an  extended  analogy  between  pure  mechanics  and  electricity.  This  extended 
analogy  is  here  briefly  outlined. 


x^vt  ( I ) 

in  which  x  is  the  distance 
traveled  in  /  seconds  by  a 
body  moving  at  velocity  v. 

W==Fx  (4) 

in  which  IV  is  the  work 
done  by  a  force  F  in  pull- 
ing a  body  through  the  dis- 
tance X. 

P^Fv  (7) 

in  which  P  is  the  power 
developed  by  a  force  F  act- 
ing upon  a  body  moving  at 
velocity  v. 

W=  yi  mv"^  ( 10 ) 
in  which  W  is  the  kinetic 
energy  of  a  mass  m  moving 
at  vlocity  ev. 


f^ui  (2) 

in  which  9  is  the  angle 
turned  in  t  seconds  by  a 
body  turning  at  angular  ve- 
locity u. 

W=T<p  (5) 

in  which  IV  is  the  work 
done  by  a  torque  7^  in  turn- 
ing a  body  through  the 
angle  1/). 

P=  Tcj  (8) 

in  which  P  is  the  power 
developed  by  a  torque  T 
acting  on  a  body  turning  at 
angular  velocity  u. 

W^yiKu"^  (II) 
in  which  IV  is  the  kinetic  en- 
ergy of  a  wheel  of  moment 
of  inertia  IC  turning  at  an- 
gular velocity  u. 


q=ii         (3) 

in  which  q  is  the  electric 
charge  which  in  t  seconds 
flows  through  a  circuit  car- 
rying a  current  ?'. 

W=Eq  (6) 

in  which  W  is  the  work 
done  by  an  e.  m.f.  E  in 
pushing  a  charge  q  through 
a  circuit. 

P=Ei  (9) 

in  which  P  is  the  power 
developed  by  an  e.  m.  f. 
E  in  pushing  a  current  i 
through  a  circuit. 

W=%Li^  (12) 
in  which  IV  is  the  kinetic 
energy  of  a  coil  of  induc- 
tance L  carrying  a  current  /. 


l6         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


F-^^  (13) 
in  which  F  is  the  force  re- 
quired to  cause  the  velocity 
of  a  body  of  mass  m  to  in- 

,  dv 

crease  at  the  rate  — - 

dt 


x^aF         (i6) 

V=r   (^9) 


G> 


y-F 


G^ 


Fig.  a.  Art.  i8. 
A  body  of  mass  in  is  sup- 
ported by  a  fiat  spring  S 
clamped  in  a  vise  as  shown 
in  Fig.  a.  A  force  F  push- 
ing sidewise  on  m  moves  it 
a  distance  x  which  is  pro- 
portional to  F  according 
to  equation  (i6).  When 
started  the  body  m  will  con- 
tinue to  vibrate  back  and 
forth  and  the  period  r  of  its 
vibrations  is  determined  by 
equation  (19). 


„,  dw 

T=K^         (14) 

in  which  T  is  the  torque 
required  to  cause  the  angu- 
lar velocity  of  a  wheel  of 
moment    of    inertia  K   to 

,  dti> 

mcrease  at  the  rate  — 

dt 


<?  =  bT 


(17) 


A-T^IC     I 


TAra 


body 


Fig.  h.  Art.  18. 
A  body  of  moment  of  in- 
ertia K  is  hung  by  a  wire 
as  shown  in  Fig.  b.  A 
torque  T  acting  on  the  body 
will  turn  the  body  and  twist 
the  wire  through  an  angle 
^  which  is  proportional  to 
T  according  to  equation 
(17).  When  started,  the 
body  will  vibrate  about  the 
wire  as  an  axis  and  the 
period  r  of  its  vibrations 
is  determioed  by  equation 
(20). 


dt 


(15) 


in  which  E  is  the  e.m.f. 
required  to  cause  a  current 
in  a  coil  of  inductance  L  to 

increase  at  the  rate  — 
dt 


q=JE 


(18) 

(21) 


I 


Fig.  c.  Art.  18. 

A  condenser  J  is  con- 
nected to  the  terminals  of  a 
coil  of  inductance  L  as 
shown  in  Fig  c.  An  e.  m. 
f.  E  acting  anywhere  in 
the  circuit  pushes  into  the 
condenser  a  charge  q  which 
is  proportional  to  E  accord- 
ing to  equation  (18).  When 
started  the  electric  charge 
will  surge  back  and  forth 
through  the  coil  constitut- 
ing what  is  called  an  oscil- 
latory current  and  the  period 
of  one  oscillation  is  deter- 
mined by  equation  (21). 


Problems. 
i      I.  The  intensity  of  the  magnetic  field  in  the  air  gap  between 
the  pole  face  and  the  armature  core  of  a  dynamo  is  5,000  c.  g.  s. 
units  and  the  pole  face  is  10  cm.  x  20  cm.      Required  the  mag- 
netic flux  from  pole  face  to  armature  core. 


INDUCTANCE   AND    CAPACITY.  1 7 

2.  A  coil  of  an  alternator  armature  has  20  turns  and  engages 
the  whole  1,500,000  lines  which  flow  from  a  pole  of  the  field 
magnet.  In  ^^-g-  second  this  coil  moves  from  a  north  pole  to  an 
adjacent  south  pole  of  the  field  magnet  when  the  flux  is  reversed. 
Calculate  the  average  e.  m.  f  in  the  coil  during  this  interval. 

3.  Find  the  approximate  inductance,  in  henrys,  of  a  cylindrical 
coil  25  cm.  long,  5  cm.  mean  diameter  wound  with  one  layer  of 
wire  containing  i  50  turns. 

4.  Calculate  the  kinetic  energy  in  joules  of  a  current  of  20 
amperes  in  the  above  coil. 

5.  The  above  coil  is  connected  to  1 10  volt  mains,  find  the  rate, 
in  amperes  per  second,  at  which  the  current  begins  to  increase  in 
the  coil. 

6.  Calculate  the  rate  at  which  the  current  is  increasing  (prob- 
lem 5)  when  it  has  reached  the  value  of  10  amperes,  the  resist- 
ance of  the  coil  being  0.25  ohm. 

7.  A  coil  of  which  the  resistance  is  2.5  ohms  and  the  induc- 
tance 0.04  henry  has  a  current  started  in  it.  The  coil  is  then 
short  circuited  and  the  current  left  to  die  away.  Calculate  the 
rate,  in  amperes  per  second,  at  which  the  current  is  decreasing  as 
it  passes  the  value  of  10  amperes. 

8.  A  coil  of  wire  has  an  inductance  of  .035  henry,  calculate 
the  magnetic  flux  through  the  coil  due  to  a  current  of  5  c.  g.  s. 
units  in  the  coil.  If  the  coil  has  1,500  turns  of  wire  calculate 
number  of  lines  of  flux  through  a  mean  turn. 

9.  A  condenser  has  a  capacity  of  1.2  microfarads,  calculate 
the  charge  which  is  pushed  into  this  condenser  by  an  e.  m.  f  of 
1,000  volts,  and  calculate  the  time  during  which  this  would  main- 
tain a  current  of  one  ampere. 

10.  The  field  coils  of  a  shunt  dynamo  have  a  resistance  of  100 
ohms  and  an  inductance  of  20  henrys.  An  e.  m.  f.  of  500  volts 
is  applied.  Calculate  the  time  required  for  the  current  to  reach 
a  value  of  4  amperes. 

2 


CHAPTER   11. 

THE  SIMPLE  ALTERNATOR. 
19.  The  alternator  consists  essentially  of  a  magnet,  near  the 
poles  of  which  a  coil  of  wire  is  moved  in  such  a  way  that  the 
magnetic  flux  from  the  poles  passes  through  the  coil,  first  in  one 
direction  and  then  in  the  other.  This  varying  magnetic  flux  in- 
duces an  electromotive  force  in  the  coil,  first  in  one  direction  and 
then  in  the  other.  This  e.  m.  f ,  called  an  altermating  e.  m.  f., 
produces  an  alternating  current  in  the  coil  and  in  the  circuit, 
which  is  connected  to  the  terminals  of  the  coil. 


Fig.  io. 
A  common  type  of  alternator  consists  of  a  multipolar  electro- 
magnet (the  field  magnet)  of  which  the  poles  project  radially  in- 

(i8) 


THE   SIMPLE  ALTERNATOR.  1 9 

ward  towards  the  passing  teeth  of  a  rotating  laminated  iron  drum 
A  (the  armature),  as  shown  in  Fig.  lo.  On  the  armature  shaft, 
at  one  end  of  the  armature,  are  mounted  two  insulated  metal 
rings  r  r  {collecting  rings),  upon  which  metal  springs  [brushes)  rub, 
keeping  continuous  contact  with  the  terminals  of  an  external  cir- 
cuit. The  ends  of  the  armature  wire  are  fastened  to  the  respec- 
tive collecting  rings,  the  armature  coils  being  wound  around  the 
teeth,  as  shown. 

The  e.  m.  f.'s  induced  in  adjacent  coils  are  in  opposite  direc- 
tions and  the  coils  are  so  connected  together  that  these  e.  m.  f 's 
do  not  oppose  each  other.  This  is  done  by  reversing  the  con- 
nections of  every  alternate  coil,  as  indicated  by  the  dotted  lines 
connecting  the  coils. 

The  field  magnet  of  the  alternator  is  excited  by  a  continuous 
current  from  some  independent  source,  generally  an  auxiliary 
dynamo  called  the  exciter.  The  type  of  armature  winding  shown 
in  Fig.  lo  is  known  as  the  concentrated  ^m6ix\^.  In  this  type  of 
winding  the  armature  conductors  are  grouped  in  a  few  heavy 
coils.  Alternator  armatures  are  also  made  in  which  the  winding 
is  distributed  in  a  large  number  of  small  slots.  This  type  of 
winding  is  known  as  the  distributed  winding ;  it  is  described  in 
Chapter  IX. 

20.  Speed  and  frequency. — The  e.  m.  f  of  an  alternator  passes 
through  a  set  of  positive  values,  while  a  given  coil  of  the  arma- 
ture is  passing  from  a  south  to  a  north  pole  of  the  field  magnet, 
and  through  a  similar  set  of  negative  values,  while  the  coil  is 
passing  from  a  north  pole  to  a  south  pole,  or  vice  versa.  The 
complete  set  of  values,  including  positive  and  negative  values, 
through  which  an  alternating  e  m.  f.  (or  alternating  current)  re- 
peatedly passes,  is  called  a  cycle.  The  number  of  cycles  per 
second  is  called  thefregtiencyf. 

'Let  p  be  the  number  of /^zri"  of  field  magnet  poles,  7i  the  revo- 
lutions per  second  of  the  armature,  and  jT  the  frequency  of  the 
e.  m.  f.  of  the  alternator.     Then 


20    THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

f=pn  (I8) 

This  is  evident  when  we  consider  that  the  e.  m.  f.  passes 
through  a  complete  cycle  of  values  while  an  armature  tooth  is 
passing  from  a  north  pole  to  the  next  north  pole,  so  there  are  p 
cycles  of  e.  m.  f.  for  each  revolution  of  the  armature.  The  fre- 
quency depends  only  on  the  speed  and  number  of  poles  and  is 
not  dependent  in  any  way  upon  the  style  of  armature  or  arma- 
ture winding. 

21.  Electromotive  force  and  current  curves. — The  successive 
instantaneous  values  of  the  e.  m.  f.  of  an  alternator  may  be  rep- 
presented  by  the  ordinates  of  points  on  a  curve,  the  abscissas 
representing  time  elapsed  from  some  chosen  epoch  ;  the  resulting 
curve  is  called  the  e.  m.  f.  curve  of  the  alternator.  In  a  similar 
manner  the  successive  instantaneous  values  of  an  alternating  cur- 
rent may  be  represented  by  ordinates  and  the  elapsed  time  by 
abscissas  giving  a  current  curve.  These  curves  are  determined 
with  the  help  of  the  contact  maker  as  explained  in  Art.  25. 
Examples:  The  full  line  curve.  Fig.  11,  represents  the  e,  m.  f. 


of  a  sniooth  core  alternator  and  the  dotted  curve  represents  the 
current  which  this  e.  m.  f.  produces  in  a  non-inductive  circuit; 
this  current  is  at  each  instant  equal  to  the  e.  m.  f.  divided  by  the 
resistance  of  the  circuit  so  that  the  current  is  a  maximum  when, 
the  e.  m.  f.  is  a  maximum,  the  current  is  said  to  be  in  phase  with 
the  e.  m.  f.  as  is  explained  in  Chapter  IV. 


THE   SIMPLE   ALTERNATOR. 


21 


The  full  line  curve,  Fig.  12,  represents  the  e.  m.  f.  of  a  smooth 
core  armature  and  the  dotted  curve  represents  the  current  which 
this  e.  m.  f.  produces  in  an  inductive  circuit.     In  this  case  part 


Fig.  12. 


of  the  e.  m.  f.  is,  at  each  instant,  used  to  cause  the  current  to 

di 
increase  or  decrease.     The  part  so  used  is  Z  -r  accordmg  to 

equation  (3),  and  the  remainder,  equal  to  Ri,  is  used  to  overcome 


Fig.  13. 

the  resistance  of  the  circuit.     When  the  current  is  zero  then  all 

the  e.  m.  f.  is  used  to  cause  the  current  to  change  since  Ri  is 

di  .  ... 

zero.     When  -r  is  zero,  the  current  is  at  its  maximum  or  muii- 
dt 


22    THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

mum  value,  and,  at  this  instant,  all  the  e.  m.  f.  is  used  to  over- 

di  . 
come  the  resistance  of  the  circuit  since  Z  —  is  zero.     It  is  to  be 

at 

particularly  noted  that  the  time  t ',  Fig.  1 2,  at  which  the  current 

reaches  its  maximum  value  is  later  than  the  time  t  at  which  the 

e.  m.  f.  reaches  its  maximum  value.      In  some  cases,  however, 

the  current  may  reach  its  maximum  value  before  the  e.  m.  f. 

The  curve,  Fig.  13,  represents  the  e.  m.  f.  of  an  alternator  with 
a  toothed  armature  core. 

22.  Instantaneous  and  average  power  expended  in  an  alternat- 
ing current  circuit. — Let  e  be  the  value  at  a  given  instant  of  an 
alternating  e.  m.  f.  and  i  the  value  of  the  current  at  the  same  in- 
stant. Then  ei  is  the  power  in  watts  which,  at  the  given  instant, 
is  being  expended  in  the  circuit,  and  the  average  value  o{  ei  is 
the  average  power  expended  in  the  circuit.     In  Fig.  14  the  full 


Fig.  14. 

line  curve  represents  the  e.  m.  f  of  an  alternator  and  the  dotted 
curve  represents  the  current  produced  by  the  alternator  in  a  cir- 
cuit having  inductance.  The  ordinates  of  the  dot  dash  curve 
represents  the  successive  instantaneous  values  of  the  power  ei. 
As  is  shown  in  the  figure,  the  power  has  both  positive  and  nega- 
tive values,  the  alternator  does  work  on  the  circuit  when  ei  is 
positive  and  the  circuit  returns  power  to  the  alternator  when  ei  is 
negative,  and,  of  course,  while  ei  is  negative  the  dynamo  is  mo- 
mentarily a  motor  and  may  for  the  moment  return  power  to  the 
fly-wheel  of  the  engine. 


THE   SIMPLE   ALTERNATOR.  23 

When  the  inductance  of  the  circuit  of  an  alternator  is  veiy 
large  the  e.  m.  f.  and  current  curves  are  as  shown  in  Fig.  15,  the 
instantaneous  power  ei  passes  through  approximately  similar  sets 
of  positive  and  negative  values  as  shown  by  the  dot  and  dash 
curves,  and  the  average  power  is  zero. 

23.  Average  values  and  effective  values. — The  average  value 
of  an  alternating  current  or  e.  m.  f  is  zero,  inasmuch  as  similar 


Fig.  15. 

sets  of  positive  and  negative  values  occur.  The  average  value  of 
an  e.  m.  f.  or  current  during  the  positive  (or  negative)  part  of  a 
cycle  is  usually  spoken  of  briefly  as  the  average  or  mean  value 
and  is  not  zero. 

Effective  values. — Consider  an  alternating  current  of  which  the 
instantaneous  value  is  i.  The  rate  at  which  heat  is  generated  in 
a  circuit  through  which  the  current  flows  is  Ri^,  where  R  is  the 
resistance  of  the  circuit,  and  the  average  rate  at  which  heat  is 
generated  in  the  circuit  is  R  multiplied  by  the  average  value  of 
2^  A  continuous  current  which  would  produce  the  same  heating 
effect  would  be  one  of  which  the  square  is  equal  to  the  average 
value  of  i^  or  of  which  the  actual  value  is  equal  to  v^  average  z^- 
This  square  root  of  the  average  square  of  an  alternating  current 
is  called  the  effective  value  of  the  alternating  current.  Similarly 
the  square  root  of  the  average  square  of  an  alternating  e.  m.  f  is 
called  the  effective  value  of  the  alternating  e.  m.  f. 

Ammeters  and  voltmeters  used  for  measuring  alter?iating  currents 
and  alternating  e.  'in.  f.^s  always  give  effective  values  as  is  shoivn  in 
Chapter  III ;  and  in  specifying  an  alternating  e.  m.  f.  or  current 
its  effective  value  is  always  used. 


24    THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 


Remark. — The  ratio  effective  value  divided  by  average  value  of 
an  alternating  e.  m.  f.,  or  current,  is  in  most  cases  approximately 


equal  to 


2  V2 


=  I.I  I  as  is  shown  in  Chapter  IV. 


Example. — Consider  the  successive  instantaneous  values  (sepa- 
rated by  equal  time  intervals)  of  an  alternating  current.  The 
sum  of  these  values  divided  by  their  number  gives  their  average 
value.  Square  each  instantaneous  value.  Add  these  squares, 
divide  by  their  number  and  extract  the  square  root,  and  the  re- 
sult is  the  square-root-of-average-square,  or  effective,  value  of 
the  current. 

24.  The  fundamental  equation  of  the  alternator. — The  equa- 
tion which  expresses  the  effective  value  of  the  e.  m.  f.  of  an 

alternator  in  terms  of  the  ar- 
mature speed  n,  the  number 
of  pairs  of  field  magnet  poles 
p,  the  flux  N  from  one  pole 
of  the  field  magnet  and  the 
total  number  of  armature 
conductors  C  which  cross  the 
face  of  the  armature  is  called 
the  fundamental  equation  of 
the  alternator.  This  equation  is  important  in  designing.  It 
is  derived  as  follows  for  the  case  in  which  the  armature  conduc- 
tors are  concentrated  in  2p  slots,  one  to  each  field  pole  as  shown 
in  Fig.  16. 

Let  N  be  the  lines  of  flux  from  one  pole,  then  one  armature 
conductor  in  one  revolution,  cuts  2pN  lines,*  and  in  one  second 
it  cuts  2pNn  lines,  which  is  the  average  e.  m.  f.  (in  c.  g.  s.  units) 
induced  in  one  armature  conductor.     We  have,  therefore, 

*  Since  we  are  concerned  with  the  average  value  during  half  a  cycle  the  change 
of  sign  during  the  two  halves  of  a  cycle  is  to  be  ignored  and  the  flux  from  north  and 
from  south  poles  is  to  be  treated  without  regard  to  sign 


Fig.   16. 


THE   SIMPLE   ALTERNATOR. 


25 


Average*  e.  m.  f.  of  alternator 


2pNCn 


\QP 


(19) 


The  ratio,  effective  e.  m.  f.  divided  by  average  e.  m.  f.,  is  for 
commercial  alternators,  approximately  equal  to  i.ii.f  There- 
fore, the  effective  e.  m.  f  of  an  alternator  with  concentrated 
armature  winding  is  approximately 


Fig.  17. 


E  = 


2.22pNCn 
10^ 


(20) 


or,  since  pn  is  the  frequency  according  to  equation  (18)  we  have 

2.22NCf 

E= ^ —  (21) 

In  which  N  is  the  magnetic  flux  from  one  pole,  and  C  is  the  total 
number  of  conductors  on  the  armature  which  are  connected  in 
series.     Sometimes  it  is  more  convenient  to  have  the  equation 

*That  is  the  average  during  half  a  cycle  as  explained  in  Art.  22.     The  average 
during  a  whole  cycle  is  zero, 
t  See  Arts.  23,  46  and  47. 


26 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


given  in  terms  of  armature  turns  instead  of  armature  conductors. 
The  formula  then  becomes 


(22) 


2"  being  the  number  of  armature  turns  in  series  between  the  col- 
lector rings, 

25,     Experimental  determination  of  alternator  e.  m.  f.  curves. 

T/ie  contact  maker. — A  disc  DD,  Figs,  i  /  and  1 8,  fixed  to  and 
rotating  with  the  armature  shaft,  carries  a  pin  p  which  makes 
momentary  electrical  contact,  once  per  revolution,  with  a  jet  of 
conducting  liquid  which  issues  from  a  nozzle  n.  This  nozzle  is 
carried  on  a  pivoted  arm  a,  and  can  be  moved  at  will,  its  position 


Fig.  li 


being  read  off  the  divided  circle  cc  One  terminal  of  an  electro- 
static voltmeter  Q  is  connected  directly  to  one  brush  of  the  alter- 
nator, while  the  other  terminal  of  the  voltmeter  is   connected 


THE    SIMPLE   ALTERNATOR. 


27 


through  the  jet  and  pin  to  the  other  brush  of  the  alternator  as 
shown  in  Fig.  18.  The  voltmeter  then  indicates  the  value  of  the 
e.  m.  f.  of  the  alternator  at  the  instant  of  contact  of  jet  and  pin. 
By  shifting  the  jet,  step  by  step,  around  the  circle  successive  in- 
stantaneous values  of  the  e.  m.  f.  may  be  determined.  The  e. 
m.  f.  passes  through  a  complete  cycle  of  values   while  the  jet  is 

shifted  -  of  a  revolution,  p   being  the   number  of  pairs  of  poles 

of  the  alternator.  In  order  that  the  e.  m.  f.  acting  upon  the  elec- 
trostatic voltmeter  may  not  fall  off  appreciably  in  the  intervals 
between  successive  contacts  of  pin  and  jet,  a  condenser /is  con- 
nected as  shown  in  Fig.  18.     The   indications  of  an  electrostatic 

voltmeter  are  not  accurate  for  small  

deflections  and  in  using  such  an  in- 
strument for  measuring  a  compara- 
tively small  e.  m.  f.  a  battery  of  known 
e.  m.  f.  may  be  connected  in  the  cir- 
cuit so  as  to  raise  the  e.  m.  f.  to  an 
accurately  measurable  value. 

In  the  determination  of  an  alternat- 
ing current  curve,  the  current  is  sent 
through  a  non-inductive  resistance 
R,  Fig.  19,  and  the  e.  m.  f.  between 
the  terminals  of  this  resistance  is  determined  as  before,  the  disc 
D  D  being  fixed  to  the  armature  shaft  of  the  alternator  which 
is  furnishing  the  current.  The  current  at  each  instant  is  equal 
to  the  e.  m.  f.  divided  by  R. 


R 


Fig.   19. 


Problems. 

1.  An  alternator  has  16  poles  and  its  speed  is  900  revolutions 
per  minute.     What  is  the  frequency  of  its  e.  m.  f.  ? 

2.  An  alternator  has  8  poles  and  its  speed  is  900  revolutions 
per  minute.  The  flux  from  one  pole  is  2,200,000  lines.  The 
armature  has    1000  conductors  (wound  in  8  slots)  all  of  which 


28         THE    ELEMENTS   OF   ALTERNATING   CURRENTS. 

are  connected  in  series.     What  will  be  the  effective  e.  m.  f.  ob- 
tained between  the  collector  rings  ? 

3.  An  alternator  has  10  poles  and  runs  at  a  speed  of  1500 
revolutions,  generating  2000  volts.  The  flux  from  one  pole  is 
2,250,000  lines.  How  many  turns  must  there  be  on  the  arma- 
ture if  they  are  all  connected  in  series  ? 

4.  The  following  are  the  instantaneous  values  of  an  e.  m.  f. 
taken  at  equal  intervals  during  half  a  cycle :  o,  30,  60,  80,  90, 
100,  90,  80,  60,  30,  o  volts.  The  corresponding  values  of  the 
current  are  —45,  —25,  o,  25,  50,  65,  75,  75,  70,  60,  45  amperes. 

Find  the  effective  value  of  the  e.  m.  f.  Find  the  instantaneous 
values  of  the  power  and  find  the  average  power. 


CHAPTER   III. 

ALTERNATING  AMMETERS,    VOLTMETERS   AND 
WATTMETERS. 

26.  The  hot  wire  ammeter  and  voltmeter.* — In  these  instru- 
ments the  current  to  be  measured  is  sent  through  a  stretched  wire. 
This  wire,  heated  by  the  current,  lengthens  and  actuates  a  pointer 
which  plays  over  a  divided  scale. 

Tlie  hot  wire  instniment,  wheii  calibrated  by  continuous  currents, 
indicates  effective  values  of  alternating  currents  ;  and  zvJien  cali- 
brated by  continuous  e.  m.f.'s  it  indicates  effective  values  of  alter- 
nating e.  m.  f.'s. 

Proof :  Consider  an  alternating  current  and  a  continuous  current  C  which  give  the 
same  reading.  These  currents  generate  heat  in  the  wire  at  the  same  average  rate.  This 
rate  is  RC^  for  the  continuous  current  and  R  X  average  ?^  for  the  alternating  current, 
i  being  the  instantaneous  value  of  the  alternating  current.  Therefore  RC^  ^=  Ry^ 
average  i^  or  C^  =  average  i^  or  C=  v   average  z'^.     Q.  E.  D. 

The  proof  for  e.  m.  f. 's  is  similar  to  this  proof  for  currents. 

Remark :  The  only  hot  wire  instrument  which  is  much  used 
is  the  Cardew  voltmeter.  Such  instruments  need  to  be  frequently 
re-calibrated,  and  are,  therefore,  not  very  satisfactory. 

27.  The  electro-dynamometer  used  as  an  ammeter. — The  essen- 
tial parts  of  the  electro-dynamometer  are  shown  in  Figs.  20  and 
21.  These  figures  show  the  arrangement  of  the  parts  in  Siemens' 
type  of  instrument.  The  coil  A  is  held  stationary  by  the  frame 
of  the  instrument  while  the  coil  B  is  mounted  at  right  angles  to 
A  and  is  hung  from  a  suspension.  This  movable  coil  is  provided 
with  flexible  or  mercury  cup  connections  a  a  and  the  current  to 
be  measured  is  sent  through   both   coils  in   series.     The  force 

*  All  voltmeters,  except  the  electrostatic  voltmeter,  are  essentially  ammeters.  That 
is,  the  e.  m.  f.  to  be  measured  produces  a  current  which  actuates  the  instrument.  The 
scale  over  which  the  pointer  plays  may  be  arranged  to  indicate  either  the  value  of  the 
current  or  the  value  of  the  e.  m.  f . 

(29) 


30        THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


action  between  the  coils  is  balanced  by  carefully  twisting  a  helical 
spring  b,  one  end  of  which  is  attached  to  the  coil  B  and  the  other 
to  the  torsion  head  c.  The  observed  angle  of  twist  necessary  to 
bring  the  swinging  coil  to  its  zero  position  is  read  off  by  means 
of  the  pointer  d  and  the  graduated  scale  e.    The  pointer  f  attached 


Top  View. 


Fig.  20. 


Fig.  21. 


to  the  coil  shows  when  it  has  been  brought  to  its  zero  position. 
The  observed  angle  of  twist  of  the  helical  spring  affords  a  measure 
of-  the  force  action  between  the  coils,  and  the  current  is  propor- 
tional to  the  square  root  of  this  angle  of  twist.  In  other  forms 
of  electro-dynamometer  the  force  action  between  the  coils  moves 
the  suspended  coil  and  causes  the  attached  pointer  to  play  over 
a  divided  scale. 

The  electro-dynamometer  when  sta7idardized  by  direct  curre7its 
indicates  effective  values  of  alteiniating  currents. 

Proof :  A  given  deflection  of  the  suspended  coil  depends  upon  a  definite  average 
or  constant  force  action  between  the  coils.     The  force  action  due  to  a  constant  current 


ALTERNATING    METERS.  3  I 

c  is  kc^  (propoitional  to  c^)  and  the  average  force  action  due  to  an  alternating  current 
is  i  X  average  i^,  so  that  if  these  currents  give  the  same  deflection  we  have  kc=^  k^ 
average  i^,  or  c^  =  average  z^,  or  .:  =  V^  average  i^.     Q.  E.  D. 

Remark :  The  electro-dynamometer  is  the  standard  instrument 
for  measuring  alternating  currents  and  it  is  always  used  in  refined 
measurements.  The  most  satisfactory  type  of  electro-dynamom- 
eter is  Kelvin's  Balance. 

28.  The  electro-dynamometer  used  as  a  voltmeter. — When  used 
as  a  voltmeter  the  coils  of  the  electro-dynamometer  are  made  of 
fine  wire,  and  an  auxiliary  non-inductive  resistance  is  usually 
connected  in  series  with  the  coils. 

When  the  inductance  of  the  electro-dynamometer  coils  is  small 
such  an  instrument,  when  calibrated  by  continuous  e.  m.f.'s,  indi- 
cates effective,  values  of  alternating  e.  in.  f  's. 

When  it  is  certain  that  the  inductance  of  an  electro  dynamom- 
eter is  negligibly  small  the  instrument  may  be  used  in  refined 
alternating  e.  m.  f  measurements. 

Inductance  error  of  the  electro-dynamo7neter  used  as  a  voltmeter. — An  electro- 
dynamometer  which  has  been  calibrated  by  continuous  electromotive  force  indicates 
less  than  the  effective  value  of  an  alternating  e.  m.  f.  The  following  discussion  of 
this  error  for  the  case  of  harmonic  e.  m.  f.  presupposes  a  knowledge  of  Chapters  IV 
and  V.  Let  j^  be  the  reading  of  an  electro-dynamometer  voltmeter  when  an  alter- 
nating e.  m.  f.  (harmonic)  of  which  the  effective  value  is  E  is  connected  to  its  ter- 
minals. That  is,  J5  is  the  continuous  e.  m.  f.  which  gives  the  same  deflection  as  E 
and  since  E  gives  the  same  deflection  as  ^  it  follows  that  the  effective  current  pro- 
duced by  E  is  equal  to  the  continuous  current  produced  by  ^  ;  that  is, 

J^  E 

in  which  R  is  the  total  resistance  of  the  instrument,  Z  its  inductance,  and  u  =  2Tr/ 
where/  is  the  frequency  of  the  alternating  e.  m.  f.  Solving  equation  (i)  for  E  we 
have 


■      E  =  l^^^^B  (.3) 

That  is,  the  reading  of  the  instrument  must  be  multiplied  by  the  factor 

V  R^  +  w2  Z2 
R 

to  give  the  true  effective  value  of  an  harmonic  alternating  e.  m.  f. 
Plunger  type  voltmeters  have  inductance  errors  also. 


32    THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

29.  The  electrostatic  voltmeter. — Two  insulated  metal  plates 
which  are  connected  to  the  terminals  of  a  battery,  or  to  any 
source  of  e.  m.  f.,  attract  each  other  with  a  force  which  is  strictly 
proportional  to  the  square  of  the  e.  m.  f  This  principle  is  ap- 
plied in  the  electrostatic  voltmeter,  which  consists  essentially  of  a 
fixed  plate  and  a  suspended  plate  to  which  a  pointer  is  attached. 
The  terminals  of  the  e.  m.  f.  to  be  measured  are  connected  to 
these  plates. 

SucIl  ail  insti'iiment,  when  calibrated  by  continuous  e.  in.  f.,  in- 
dicates effective  values  of  alternating  e.  in.  f. 

Proof:  A  given  deflection  of  the  suspended  plate  depends  upon  a  definite  average 
or  constant  force  action  between  the  plates.  The  force  action  due  to  a  constant  e. 
m.  f.  ^  is  K]B^^  (proportional  to  J^^)  and  the  average  force  action  due  to  an  alter- 
nating e.  m.  f.  e,  is  -^X  average  e~.  If  these  e.  m.  f.'s  give  equal  deflections  the  force 
KJBi^  is  equal  to  the  average  force  A'X  average  e"^  so  that  J^^  =  average  e"^,  or  J^  = 
1/ average  c^.     Q.  E.  D. 

The  electrostatic  voltmeter  is  the  standard  instrument  for  mea- 
suring alternating  e,  m.  f.'s,  especially  for  the  measurement  of 


atr  air 

B  B 

Fig.  22. 

very  high  e.  m,  f.  Further,  with  high  e.  m.  f 's  the  electrostatic 
attraction  of  parallel  metal  plates  is  great  enough  to  be  accurately 
measured  by  a  balance  and  in  this  case  the.  e.  m.  f.  between  the 
plates  (constant  e.  m.  f ,  or  effective  value  of  an  alternating  e.  m. 
f.)  may  be  calculated  independently  of  calibrations  of  any  kind. 
An  instrument  arranged  for  the  absolute*  measurement  of  e.  m. 
f.  in  this  way  is  called  an  absolute  electrometer. 

*  That  is,  the  measurement  in  terms  of  mechanical  units  of  force,  distance,  etc. 


ALTERNATING   METERS.  33 

The  absolute  electrometer  consists  of  two  parallel  metal  plates, 
AaA  and  BB,  Fig.  22.  The  central  portion  a  of  the  upper 
plate,  while  remaining  in  electrical  communication  with  AA,  is 
detached  and  suspended  from  one  arm  of  a  balance  beam  as 
shown.  The  e.  m.  f.  E  between  AaA  and  BB  is  given  by  the 
formula 

E^  =  -^—  (24) 

in  which  F  is  the  observed  downward  pull  on  a  in  dynes,  d  is  the 
distance  apart  of  the  plates  in  centimeters  and  a  is  the .  area  in 
ci')^  of  the  detached  portion  a. 

30.  The  spark  gauge. — The  high  e.  m.  f.'s  used  in  break-down 
tests  are  usually  measured  by  means  of  the  spark  gauge.  This 
consists  of  an  adjustable  air  gap  which  is  adjusted  until  the  e.  m. 
f.  to  be  measured  is  just  able  to  strike  across  in  the  form  of  a 
spark.  The  e.  m.  f  is  then  taken  from  empirical  tables  based 
upon  previous  measurements  of  the  e.  m.  f.  required  to  strike 
across  various  widths  of  gap.  In  the  spark  gauge  of  the  Gen- 
eral Electric  Co.  the  spark  gap  is  between  metal  points,  one  of 
which  is  attached  to  a  micrometer  screw  by  means  of  which  the 
gap  space  maybe  adjusted  and  measured.  The  striking  distance 
in  any  spark  gauge  varies  greatly  with  the  condition  of  the 
points.  It  is,  therefore,  necessary  to  see  that  the  points  are  well 
polished  before  taking  measurements. 

31.  Plunger  type  ammeters  and  voltmeters. — In  instruments 
of  this  type  the  current  to  be  measured  passes  through  a  coil  of 
wire  which  magnetizes  and  attracts  a  movable  piece  of  soft  iron 
to  which  the  pointer  is  fixed. 

A  plunger  meter  (ammeter  or  voltmeter)  should  be  calibrated 
under  the  conditions  in  which  it  is  to  be  used.  Thus,  if  a  plunger 
instrument  is  to  be  used  as  an  ammeter  for  alternating  currents  of 
a  given  frequency  it  should  be  calibrated  by  currents  of  this 
frequency,  these  currents  being,  for  the  purpose  of  the  calibra- 
tion, measured  by  a  standard  alternating  current  ammeter  such 

3 


34 


THE  ELEMENTS  OF  ALTERNATING  CTRRENTS. 


as  an  electrodynamometer.  The  indications  of  a  plunger  instru- 
ment do  not,  however,  vary  greatly  with  frequency  and  such  in- 
struments are  used  for  approximate  measurements  without  regard 
to  frequency. 

The  Thomson  inclined  coil  meter  of  the  General  Electric  Co.  is 
of  the  plunger  type.  The  essential  parts  of  this  instrument  are 
shown  in  Fig.  23.     A  coil  A,  through  which  flows  the  current 


.    rq^^ 


Fig.  23. 

to  be  measured,  is  mounted  with  its  axis  inclined  as  shown,  A 
vertical  spindle  mounted  in  jeweled  bearings  and  controlled  by  a 
hair-spring  passes  through  the  coil,  and  to  this  spindle  are  fixed 
a  pointer  b  and  a  vane  of  thin  sheet-iron  a.  This  vane  of  iron 
is  mounted  obhquely  to  the  spindle.  When  the  pointer  is  at  the 
zero  point  of  the  scale  the  iron  vane  a  lies  nearly  across  the  axis 
of  the  coil,  and  when  a  current  passes  through  the  coil  the  vane 
tends  to  turn  until  it  is  parallel  to  the  axis  of  the  coil,  thus  turn- 
ing the  spindle  and  moving  the  attached  pointer  over  the  cali- 
brated scale. 

32.  The  potential  method  for  measuring  alternating  current. 
— The  alternating  current  to  be  measured  is  passed  through  a 
known  non-inductive  resistance  R  and  the  e.  m.  f.  between  the 
terminals  of  this  resistance  is  measured  by  a  voltmeter.  The 
current  (effective  value)  is  then  equal  to  the  e.  m.  f.  (effective 
value)  divided  by  the  resistance. 

33.  The  calorimetric  method  for  measuring  alternating  current. 
— The  current  to  be  measured  is  passed  through  a  known  resist- 


ALTERNATING  METERS. 


35 


ance  which  is  submerged  in  a  calorimeter  by  means  of  which  the 
heat  H  which  is  generated  in  the  resistance  in  an  observed  in- 
terval of  time  t  is  determined.  This  heat  being  expressed  in 
joules  we  have 

H=  PRt  (25) 

in  which  /is  the  effective  value  of  the  current. 

Measurements  of  Power  in  Alternating  Circuits.* 
34.  The  three -voltmeter  method. — A  non-inductive  resistance 
Ry  Fig.  24,  is  connected  in  series  with  the  circuit  be  in  which  the 
power  P,  to  be   determined,  is  expended. 
The  e.  m.  f 's  E^  between  ab,  E^  between 
be,  and   E^  between    ac,  are    observed  by 
means    of    a   voltmeter    as    nearly   simul- 
taneously   as   possible.     Then 


mdin 


E^ 


772  _    /72 


2R 


(26) 


Proof:  Let  e^,  e^  and  e^  be  the  instantaneous  e.  m.  f.'s 
between  ab,  be  and  ac  respectively,  then 

^3  =  ^i  +  '?2  (i) 

or  f 32  =  e^^  -f-  2^i^2  +  '?2^  ( ii ) 

or 


mtmn 


Fig.  24. 
Average  e^'^  =  average  e^^  +  2  average  e^e^  -j-  average  e^^f  (iii) 

but  £^^  =  average  e^^,  E-^-  ^  average  e.^,  and  E^  =  average  e^^.     Further  -k   is  the 

,     ^1  .     ,      . 

instantaneous  current  m  abc,  -^  •  ^^  is  the  instantaneous  povi^er  expended  in  be  and 

average  (-^-e^  j  or  —  X  average   (e^e^)  is  the  average   power  E  expended  in  be 


so  that  average  [e-^e^)=^  RE.     Therefore  equation  (iii)  becomes 
E.2  =  E^^-\-2RE+E^^ 

£2 £2 £2 


E: 


2.R 


(iv) 
Q.  E.  D. 


*  In  alternating  circuits  power  cannot  be  measured  by  means  of  an  ammeter  and 
a  voltmeter  as  in  the  case  of  direct  current  for  the  reason  that  the  power  expended  is 
in  general  less  than  the  product  of  effective  e.  m.  f.  into  effective  current  on  account 
of  the  difference  in  phase  of  the  cun-ent  and  e.  m.  f. 

fFor  proof  of  (iii)  see  proposition  Art.  45. 


36 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


35.  The  three-ammeter  method  for  measuring  power. — The  cir- 
cuit CCy  Fig.  25,  in  which  the  power  P,  to  be  measured  is  ex- 
pended, is  connected  in  parallel  with 
a  non-inductive  resistance  R  and 
three  ammeters  are  placed  as  shown. 
Then 

P=\{h'-h'-I^)     (27) 

in  which  /^,  I^  and  I^  are  the  cur- 
rents indicated  by  the  three  amme- 
ters. 

Proof:  Let  i-^,  i^  and  i^  be  the  instantane- 
ous values  of  the  currents  /j,  I^  and  I^.     Then 

or  ^32=^•l2-^-2^•l^•2  +  7■22 


(i) 
(") 


Fig.  25. 

or 

or  average  z./  =  average  ij^  -{-  2  average  (z'j  Zg)  -{-  average  Z2'^- 

But  Ii^  =  average  i-^^,     I,^  =  average  i^,     and  73^  =  average  i^. 

Further,  the  instantaneous  e.  m.  f.  between  the  terminals  of  R  or  of  CC  is  Ri^  so  that 
Ri^i-^  is  the  instantaneous  power  expended  in  CCand  ^  X  ^-verage  {i^'^^'i)  i^  ^'^  ^^" 

P 

erage  power  P,  expended  in  CC.     Therefore,  average  {i-^,  i^)  =  -^    and    equation 

(iii)  becomes : 


2P- 


(iv) 


Q.  E.  D. 


Combination  method. — The  three-ammeter  method  for  meas- 
uring power  may  be  modified  by  using  the  potential  method  for 
measuring  /g,  Fig.  25.  In  this  case  the  e.  m.  f.  between  the  ter- 
minals of  R,  Fig.  25,  is  measured  by  means  of  a  voltmeter  so  that 

'^~  R  ^2 


/„  =  —  where  E^  is  the  voltmeter  reading. 


36.  The  Wattmeter. — The  Wattmeter  is  an  electrodynamometer 
of  which  one  coil  a,  Fig.  26,  made  of  fine  wire,  is  connected  to 
the  terminals  of  the  circuit  CC  in  which  the  power  to  be  meas- 


ALTERNATING   METERS. 


37 


7naiJ% 


Fig.  26. 


ured  is  expended.     The  other  coil  b  made  of  large  wire  is  con- 
nected in  series  with  CC  as  shown.     The  fine  wire  coil  a  is  mov- 
able and  carries  the  pointer  which  in- 
dicates the  watts  expended  in  CC. 

Such  an  instrument  when  calibrated 
with  continuous  current  and  e.  m.  f.  in- 
dicates power  accurately  when  used 
with  alternating  currents,  provided  the 
inductance  of  the  coil  a  together  with 
the  auxiliary  resistance  r  is  small. '^ 

Proof. — A  given  deflection  of  the  movable  coil 
a  depends  upon  a  certain  average  or  constant  force 
action  between  the  coils.     Consider  a  continuous 

M  . 

e.  m.  f.  J^  which  produces  a  current  —  in  a  and  a  current  C  in  CC  and  <5.     The 

force  action  between  the  coils  is  proportional  to  the  product  of  the  currents  in  a 

and  (5,  that  is,  the  force  action  is  /^.  — .  C,  where  i  is  a  constant. 

r 

Consider  an  alternating  e.  m.  f.  of  which  the  instantaneous  value  is  e  ;  this  pro- 
duces  a  current  -  through  a  ( provided  the  inductance  of  a  is  zero )  and  a  current  i  in 

CC  and  b.     The  instantaneous  force  action  between  the  coils  xs  k.—  .  i  and  the  aver- 

r 

.    k 
age  force  action  is  —  .  average  (^?).     If  this  alternating  e.  m.  f.  gives  the  same  de- 
flection as  the  continuous  e.  m.  f.  then 

k  k 

-   X  average  (^?)  =  ^C  - 

or  average  {ei)  ==^C. 

That  is  the  given  deflection  indicates  the  same  power  whether  the  currents  are  al- 
ternating or  direct.  Q.  E.  D. 

Remark:  A  good  wattmeter  is  the  standard  instrument  for 
measuring  power  in  alternating  current  circuits.  The  three-am- 
meter and  the  three-voltmeter  methods  are  troublesome  and  slight 
errors  of  observation  may  in  some  cases  lead  to  very  great  errors 
in  the  result. 


*  Small,  that  is  in  comparison  with  — -  ;  where  r  is  the  total  resistance  of  a  and 


r.  Fig.  26,  and /is  the  frequency  of  the  alternating  current. 


38 


THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 


37.  The  recording  wattmeter  is  an  instrument  for  summing  up 
the  total  work  or  energy  expended  in  a  circuit. 

The  Thomson  recording  zvattmeter  is  a  small  electric  motor 
without  iron,  the  field  and  armature  coils  of  which  constitute  an 
electrodynamometer.     The  field  coils  B  of  this  motor,  Fig.  27, 

are  connected  in  series 
with  the  circuit  CC  in 
which  the  work  to  be 
measured  is  expended. 
The  armature  A,  to- 
gether with  an  auxiliary 
non-inductive  resistance 
R,  is  connected  between 
the  terminals  of  the  cir- 
cuit CC,  as  shown.  Cur- 
rent is  led  into  the  ar- 
mature by  means  of  the 
brushes  dd  pressing  on 
a  small  silver  commuta- 
tor e. 

Discussion  of  the  Thomson  recording  wattmeter. — The  driving  torque,  acting  upon 
the  armature  is  proportional  to  the  rate  at  which  work  is  spent  in  the  circuit  CC  [i.  e., 
to  the  power  expended,  as  explained  in  Art.  36).  The  instrument  is  so  constructed 
that  the  speed  of  the  armature  is  proportional  to  this  driving  torque  or  to  the  power 
spent  in  CC.  That  is,  the  rate  of  turning  of  the  armature  is  proportional  to  the  rate 
at  which  work  is  done  in  the  circuit  CC,  so  that  the  total  number  of  revolutions  turned 
by  the  armature  is  proportional  to  the  total  work  expended  in  the  circuit  CC. 

To  make  the  armature  speed  proportional  to  the  driving  torque  the  armature  is 
mounted  so  as  to  be  as  nearly  as  possible  free  from  ordinary  friction  and  a  copper 
disk/,  Fig.  27,  is  mounted  on  the  armature  spindle  so  as  to  rotate  between  the  poles 
of  peiTnanent  steel  magnets  MAI.  To  drive  such  a  disk  requires  a  driving  torque 
proportional  to  its  speed. 


Fig.  27. 


CHAPTER   IV. 

HARMONIC    ELECTROMOTIVE   FORCE  AND    CURRENT. 

38.  Definition  of  harmonic  e.  m.  f.  and  current. — A  line  OP, 
Fig.  28,  rotates  at  a  uniform  rate,  /  revolutions  per  second,  about 

a  point  O  in  the  direction  of  the  arrow 
gJi.  Consider  the  projection  Ob  of  this 
rotating  line  upon  the  fixed  line  AB ; 
this  projection  being  considered  positive 
when  above  0  and  negative  when  below 
O.  An  harmonic  e.  in.  f.  {or  atrrent)  is 
an  e.  in.  f.  ivJiich  is  at  each  instant  pro- 
portional to  the  line  Ob,  Fig.  28.  The 
line  Ob  represents  at  each  instant  the 
actual  value  e  of  the  harmonic  e.  m.  f. 
to  a  definite  scale ;  and  the  length  of  the 
line  OP  (which  is  the  maximum  length 
of  Ob)  represents  the  maximum  value  M  of  the  harmonic  e.  m.  f. 
to  the  same  scale.  The  line  Ob  passes  through  a  complete  cycle 
of  values  during  one  revolution  of  OP  and  so 
also  does  the  harmonic  e.  m.  f.  e.  Therefore  the 
revolutions  per  second  f,  of  the  line  OP  is  the 
frequency  of  the  harmonic  e.  m.  f.  e.  The  ro- 
tating hues  ^  and  I,  Fig.  29,  of  which  the  pro- 
jections on  a  fixed  line  (not  shown  in  the  figure) 
represent  the  actual  instantaneous  values  e  and 
/  if  an  harmonic  e.  m.  f.  and  an  harmonic  current  are  said  to  rep- 
resent the  harmonic  e.  m.  f.  and  current  respectively.  Of  course, 
the  rotation  of  the  lines  M  and  J  is  a  thing  merely  to  be  imagined. 

39.  Algebraic  expression  of  harmonic  e.  m.  f.  and  current. — The 
line  OP,  Fig.  28,  makes  /  revolutions  per  second  and,  therefore, 

(39) 


40    THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

it  turns  through  271/"  radians  per  second,  since  there  are  27r  radians 
in  a  revolution,  that  is 

CO  =  27tf  (28) 

in  which  co  is  the  angular  velocity  of  the  line  OP  in  radians  per 
second.  Let  time  be  reckoned  from  the  instant  that  OP  coin- 
cides with  Oa,  then  after  i  seconds  OP  will  have  turned  through 
the  angle  ^  ^  tot  and  from  Fig.  28  we  have 

Od  =  OP  sin  1^=  OP  sin  wt 
But  03  represents  the  actual  value  e  of  the  harmonic  e.  m.  f.  at 
the  time  /  and  OP  represent  its  maximum  value  M,  therefore 

£>  =  ^  sin  coi  (29) 

is  an  algebraic  expression  for  the  actual  value  e  of  an  harmonic 

e.  m.  f  at  .time  / ;  M  being-  the  maximum  value  of  e,  and  — 
being  the  frequency  according  to  equation  (28). 

Similarly  t  =  I  sin  cot  (30) 

is  an  algebraic  expression  for  the  actual  value  i  of  an  harmonic 
current  at  time  t;  I  being  the  maximum  value  of  z. 

Remark   i:     If  time  is   reckoned  from   the  instant   that    OP, 

Fig.  28,  coincides  with  the  line  Ob  then  equations  (29)  and  (30) 

become 

e  =  M  cos  cot 

i  =  I  cos  cot 

Remark  2 :  The  curve  which  represents  an  harmonic  e.  m.  f. 
or  an  harmonic  current  (see  Art.  21)  in  a  curve  of  sines. 

Remark  j :  A  great  many  alternators,  especially  those  with 
distributed  armature  windings,  generate  e.  m.  f.'s  which  are  very 
nearly  harmonic.  Calculations  in  connection  with  the  design  of 
alternating  current  apparatus  are  simple  enough  to  be  practicable 
only  when  the  e.  m.  f 's  and  currents  are  assumed  to  be  har- 
monic. Hereafter,  then,  when  speaking  of  alternating  e.  m.  f.'s 
or  currents  it  will  be  understood  that  they  are  harmonic  unless  it 
is  specified  to  the  contrary. 


HARMONIC   E.  M.  F.  AND    CURRENT.  4 1 

40.  Definitions.*  Cycle. — A  cycle  is  one  complete  set  of  values 
(positive  and  negative)  through  which  an  e.  m.  f.  or  current  re- 
peatedly passes.  The  frequency  is  the  number  of  cycles  passed 
through  per  second.  The  period  is  the  duration  of  one  cycle. 
For  example,  an  alternator  generates  e.  m.  f.  at  a  frequency  of  60 
cycles  per  second  ;  the  period  is  -^-^  of  a  second  and  the  angular 
velocity  of  the  line  OP,  Fig.  28,  is  60  revolutions  per  second  or 
\20  7t  radians  per  second  (=«>)• 

Synchronism. — Two  alternating  e.  m.  f.'s  or  currents  are  said 
to  be  in  synchronism  when  they  have  the  same  frequency.  Two 
alternators  are  said  to  run  in  synchronism  when  their  e.  m.  f.'s 
are  in  synchronism. 

41.  PliajD  difference. — Consider  two  synchronous  harmonic 
e.  m.  f.'s  e^  and  e^.  Suppose  that  e^  passes  through  its  maximum 
value  before  e^ ;  then  e^   and   e^  are   said  to 

differ  in  phase.     The  line  ^j,  Fig.  30,  which 

represents  e^  must  be  ahead  of  the    line  H^ 

which  represents  e^  as  shown  in  the  figure. 

The  angle  d  is  called  the  phase  difference  of  e^ 

and  e^.     The  lines  'E^  and  Hg  are  supposed 

to  be  rotating  about  (9  in  a  counter-clockwise 

direction  as  explained  in  Art.  38. 

When  the  angle  d,  Fig.  30,  is  zero,  as  shown  in  Fig.  31,  the 

e.  m.  f.'s  e^  and  e^  are  said  to  be  in  phase.  In  this  case  the  e.  m. 
f.'s  increase  together  and  decrease  together; 
that  is  when  e^  is  zero  c^  is  also  zero,  when 
e^  is  at  its  maximum  value  so  also  is  e^,  etc. 

When  d  =  90°  as  shown  in  Fig.  32  the  two 
e.  m.  f.'s  are  said  to  be  in  quadrature.  In  this 
case  one  e.  m.  f.  is  zero  when  the  other  is  a 
maximum,  etc. 

*  The  definitions  of  cycle  and  frequency  given  in  Art.  20  are  here  repeated  for  the 
sake  of  clearness.  All  definitions  given  in  this  article  apply  to  alternating  currents 
and  electromotive  forces  of  any  character  as  well  as  to  harmonic  e.  m.  f.'s  and  currents. 


42 


THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 


When  6  =  i8o°  as  shown  in  Fig.  33  the  two  e.  m,  f.'s  are  said 
to  be  in  opposition.     In  this  case  the  two  e.  m.  f.'s  are  at  each  in- 
stant opposite  in  sign  and  when  one  is  at  its 
positive  maximum  the  other  is  at  its  negative 
maximum,  etc. 

42.  Composition  and  resolution  of  harmonic 
e.  m.  f.'s  and  currents,  (a)  Composition. — Con- 
sider two  synchronous  harmonic  e.  m.  f.'s  e^ 
and  ^2  represented  by  the  Hnes  ^^  and  M^,  Fig. 
34.  The  sum  e^  -f  e^  is  an  harmonic  e.  m.  f. 
of  the  same  frequency  and  it  is  represented 
by  the  hne  E).  This  is  evident  when  we 
■    consider  that  the  projection  on  any  hne  of  the 

diagonal  of  a  parallelogram  is  equal  to  the  sum  of  the  projections 

of  the  sides  of  the  parallelogram. 


^1 


ti 


Corollary. — The  sum  of 
any  number  of  synchronous 
e.  m.  f 's  (or  currents)  is  an- 
other e.  m.  f  (or  current)  of 
the  same  frequency  which  is 
represented  in  phase  and 
magnitude  by  the  line  which  is  the  vector  sum  of  the  hnes  which 
represent  the  given  e.  m.  f.'s  (or  currents).     Thus  the  lines  ^^ 

^2  and  J^g,   Fig.   35,  represent 


Fig.  33. 


———>''?' 


three  given  synchronous  har- 
monic e.  m.  f's  and  the  line 
^  (the  vector  sum  of  ^^  ^2 
and  'H^  represents  an  harmonic 
e.  m.  £  which  is  the  sum  of  the 
given  e.  m.  f's. 

{b^  Resolution. — A  given  har- 
monic e.  m.  f  (or  current)  may  be  broken  up  into  a  number  of 
harmonic  parts  of  the  same  frequency  by  reversing  the  process 
of  composition.     For  example,  the  line  ^,  Fig.   35,  represents 


Fig.  34. 


HARMONIC   E.  M.   F.  AND  CURRENT. 


43 


a  given  harmonic  e.  m.  f.  which  may  be  split  up  into  the  three 
e.  m.  f.'s  represented  by  the  Hnes  J^j,  ^^  ^^^  -^s- 
43.  Examples  of  composition  and  resolution. 

(a)     Two   alternators  A  and  B  running  in  synchronism  are 
connected  in  series  between  the  mains  as  shown  in  Fig.  36.     If 


TaaiJi 


■Bq 


"   jnam 


¥\G.  35. 


Fig.  36. 


the  e.  m.  f.'s  of  A  and  B  are  in  phase  the  e.  m.  f.  between  the 
mains  will  be  simply  the  numerical  sum  of  the  e.  m.  f.'s  oi  A  and 
B.  If,  on  the  other  hand,  the 
e.  m.  f.'s  of  A  and  B  differ  in 
phase  the  state  of  affairs  will  be 
such  as  is  represented  in  Fig.  37  ; 
in  which  the  lines  A  and  B  repre- 
sent the  e.  m.  f.'s  of  the  alterna- 


FiG.   37. 


mam 


tors  A  and  B  respectively,  Q  is  the  phase  difference  of  A  and  B, 

and  the  line  ^  represents 
the  e.  m.  f.  between  the 
mains. 

{B)  Two  alternators  A  and 
B  running  in  synchronism 
are  connected  in  parallel 
between  the  mains   shown 


Fig.  38. 


in  Ficr.  ^8.      Let  the  lines 


A  and  B,  Fig.  39,  represent  the  currents  given  by  A  and  B  re- 


44 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


spectively,  the  phase  difference  being  d  ;  then  the  current  in  the 
main  hne  is  represented  by  I. 

{c)  Two  circuits  A  and  B  are  connected  in  series  between  the 
mains  of  an  alternator  as  shown  in  Fig.  40.     The  Hne  J^,  Fig.  41, 

represents  the  e.  m.  f.  between 
the  mains,  the  Hne  A  repre- 
sents the  e.  m.  f.  between  the 
terminals  of  the  circuit  A  and 
the  line  B  represents  the  e.  m. 
f.  between  the  terminals  of  the 
circuit  B.  If  the  circuits  A  and  B  have  inductance  it  may  be 
that  the  e.  m.  f.  A  and  the  e.  m.  f.  B  are  not  in  phase  with  each 

other  in  which  case  the  relation 
between  A,  B  and  E  will  be  as 
shown  in  Fig.  41.  If  one  of  the 
circuits  A  or  B  contains  a  con- 


juain 


jraam 


Fig.  40. 

denser  then  the  e.  m.  f.'s  A  and  B,  Fig.  41,  may  be  nearly  op- 
posite in  phase,  and  A  and  B  may  each  be  indefinitely  greater 
than  the  e.  m.  f.  E  between  the  mains, 

{d)  Two  circuits  A  and  B,  Fig.  42,  are  connected  in  parallel 
across  the  terminals  of  an  alternator  as  shown.  The  current  / 
from  the  alternator  is  related  to  the  currents  A  and  B  as  shown 
in  Fig.  43.  If  one  of  the  circuits  A  or  B  contains  a  condenser 
then  the  currents  A  and  B  may  be  nearly  opposite  in  phase  and 
the  currents  A  and  B  may  each  be  indefinitely  greater  than  the 
current  /from  the  alternator. 

44.  Rate  of  change  of  harmonic  e.  m.  f 's  and  currents. — Con- 
sider the  harmonic  current  [see  equation  (30)]: 


HARMONIC   E,  M.  F.  AND   CURRENT. 


45 


i  =  I  sin  (tii 


(^) 


jxiain 


When  this  current  is  sent  through  an  inductive  circuit  an  e.  m.  f. 
Z^  -T  is  at  each  instant  required  to  make  the  current  increase  or 

decrease.  In  the  study  of  alternat- 
ing currents  in  inductive  circuits  it  is, 
therefore,  necessary  to  consider  the 

r      ^  ^^'        r       1 

rate  of  change  -y  of  the  current. 

Differentiating  the  above  expres- 
sion for  i  with  respect  to  time  we 
have 


dz 

—r  =  col  COS    cot 

at 


(b) 


or 


di 
~dt 


=  wXsin  {cot  -f-  90) 


Fig.  42. 


di 


(31) 


This  equation  shows  that  the  rate  of  cliange,  -j-  of  an  harmo- 
nic current  may  be  represented  by  the  projection*  of  the  hne  wj, 


_^^_    _    fixed  line . 
Fig.  44. 

Fig.  44,  which  is  90°  ahead  of  the  hne  I  which  represents  the 
current. 

di 
The  relation  of  i  and  -j-  is  most  clearly  shown  by  the  sine 

curve  diagram.     Thus  the  full  Hne  curve.  Fig.  45,  represents  the 

harmonic  current  i.     The  steepness  of  this  curve  at  each  point 

di 
represents  the  value  of  -7- .     The  steepness  of  this  curve  is  great- 

*  On  a  vertical  fixed  line  not  shown  in  the  figure. 


46         THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 


est  at  the  point  a  where  the  curve  crosses  the  axis,  hence  the  value 

of  -7  is  a  maximum  90°  before  i  reaches  its  maximum.    The  or- 
dt 

di 
dinates  of  the  dotted  curve  represent  the  values  of  -7  - 


Fig.  45. 


di 


til 

Remark :  It  is  to  be  noted  that  the  portion,  L  -r  ,  of  the  total 

e.  m.  f.  acting  on  the  circuit,  which  is  used  to  cause  the  current 
to  increase  and  decrease,  is  represented  by  the  line  (oLI,  Fig.  46  : 
the  line  I  represents  the  current  in  the  circuit. 

45.  Average  or  mean  value  of 
an  harmonic  e.  m.  f.  or  current. — 
Consider   any  varying   quantity 


Fig.  46.  Fig.  47. 

jF.  Its  average  value  during  an  interval  of  time  from  t'  to  f  is 
-jj^ — J  the  summation  being  extended  throughout  the  interval. 
That  is, 


HARMONIC   E.  M.  F.  AND    CURRENT.  47 

If  the  successive  values  of  y  be  represented  by  the  ordinates 

of  the  curve,  Fig.  47,  and  the  corresponding  values  of  the  time  / 

C'" 
be  represented  by  the  abscissas,  then  I  ydt  is  the  area    of    the 

I        c^" 
shaded  portion  and  -77 -,  |  ydt  is  the    height  of  a   rectangle 

t'cdt"  of  the  same  area  as  the  curve  and  having  the  same  base. 
The  average  ordinate  of  such  a  curve,  as  Fig.  47,  during  a  given 
interval  of  time  may  be  obtained  quite  closely  by  measuring  the 
lengths  of  a  number  of  equidistant  ordinates.  The  sum  of  these 
ordinates  divided  by  the  number  of  ordinates  gives  the  average 
ordinate  of  the  curve. 

Fi'oposition  :     The  average  value  of  the  sum  of  a  number  of  quantities  is  equal  to 
the  sum  of  the  average  values  of  each. 

Proof:  Let  x,y,  z,  ...  be  the  quantities.     Then  by  definition  we  have 

but 

=^  Av.  x-\- Az'.  y -\- ...  (ii) 

Therefore 

Av.  {x-\-y-\-z-\- ...)=Av   x-\- Av.  y-\'Av.z-\- ...  (33) 

Q.  E.  D. 

46.  Proposition. — The  average  value  of  an  harmonic  e.  m.  f  or 

current  during  half  a  cycle  Uot  z=  o  to  (ot  =  n ;  or  t  =  o  io  t  =  -\ 

2  maximum  value 
is 

TZ 
Proof:   Let  ^  =  J^  sin  ut  be  the  harmonic  e.  m.  f.     Substitute  J5  sin  u/  for  y  in 
equation  (32)  and  we  have 


Av.  I?  =  7, ^  I     €m.Udt. 


ft" 
4   ""^ 


Substituting  x  for  ut  and  remembering  that  the  limits  are  from  wi'  =  o  to  w/  =  tt  we 
have 

A\.e  =  —  I     sinxdx=: —       cos  x  -  = (34)  Q.  E.  D. 

TT  Jo  TT   Lo  TT 

or  the  average  value  of  the  harmonic  e.  m.  f.  is  twice  the  maximum  value  divided  by 


48    THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

2 

TT.    Since  -  =  .636  it  may  also  be  stated  that  the  average  value  of  an  harmonic  e.  m.  f., 

TT 

or  current,  is  .636  times  the  maximum  value. 

Remark :  The  average  value  of  an  harmonic  e.  m.  f.  or  current 
during  one  or  more  whole  cycles  is  zero, 

47.  Proposition. — The  square  root  of  mean  square,  or  effective 

value,  of  an  alternating  e.  m.  f  or  current  during  one  or  more 

-    .                     maximum  value 
whole  cycles  is  equal  to  ■ ;= or  .707  x  maximum. 

V2 

Proof:  Let  ^  =  ^  sin  ut  be  a  harmonic  e.  m.  f.  To  find  the  average  value  of 
(f2z=  Ri  sin  2  ut  it  is  necessary  to  find  the  average  value  of  the  square  of  the  sine  of 
the  uniformly  variable  angle  w^.     We  have  the  general  relation 

sin2  idt  -\-  cos2  ut^i  '  (a) 

so  that  by  equation  (33) 

Av.  sin2  u(  -\-  Av.  cos^  w^=  i.  (b) 

Now,  the  cosine  of  a  uniformly  variable  angle  passes  similarly  through  the  same  set 
of  values  during  a  cycle  as  the  sine,  hence  Av.  sin^  ut  and  Av.  cos^  ut  are  equal,  so 
that  from  (b)  we  have  : 

2  Av.  sin^  ut=i 

or  Av.sm^ut  =  }i. 

The  average  value  of  e^  is 

Av.  ,»2  =  J^2  Av.sin2(irf 

Av.  e^=^=^— 
2 

J^ 

and  t/Av.  f2=_  (35) 

Q.  E.  D. 

Note :  The  square  root  of  mean  square  value  of  an  harmonic 
e.  m.  f.  or  current  is  often  spoken  of  as  the  effective  value  of  the 
e.  m.  f.  or  current.  When  it  is  stated  that  an  alternating  current 
is  so  many  amperes,  the  effective  or  square  root  of  mean  square 
value  is  always  meant.  The  same  is  also  true  with  regard  to  al- 
ternating e.  m.  f.'s.  Hereafter  the  symbol  E  will  be  used  to 
designate  the  effective  value  of  an  e.  m.  f.  and  /  the  effective 
value  of  an  alternating  current.  In  case  the  currents  and  e.  in. 
f.'s  are  harmonic  we  have  the  relations 


HARMONIC   E.  M.  F.  AND    CURRENT.  49 

£=^  (36) 

in  which  ^  and  J  are  the  maximum  values  of  the  e.  m.  f,  and 

current  respectively. 

.     effective  value 

Note :  The  ratio  -. —  of  an  alternating-  e.  m.  f.  or  cur- 

average  value 

rent  is  sometimes  spoken  of  as  thQforni  factor  of  the  e.  m.  f.  or 

current,  because  this  ratio  depends  upon  the  shape  or  form  of  e. 

m.  f.  or  current  curve.     The  form  factor  in   the  case    of  har- 

•  707 
monic  e.  m.  f.'s  or  currents  would  be    ,    ,  =  i.ii. 

.636 

48.  Power. — As  pointed  out  in  Article  22,  Chapter  II.,  the 
power  developed  by  an  alternating  e.  m.  f.  pulsates  and  in  most 
practical  problems  it  is  the  average  power  de- 
veloped which  is  the  important  consideration. 
Let  e  =  H  sin  cot  be  an  harmonic  e.  m.  f. 
acting  on  a  circuit  and  i  =  I  sin  {cot  —  d)  the 
current  produced  in  the  circuit ;  d  being  the 
difference  in  phase  of  the  e.  m.  f.  and  current 
as  shown  in  Fig.  48.  The  power  developed  at  a  given  instant 
is  ei  and  in  order  to  estimate  the  average  power  developed  we 
must  find  an  expression  for  the  average  value  of  ei.     We  have 

ei  =  MI  sin  cot  sin  (cot  —  d) 
or  since  sin  (^cot  —  d)  =  sin  cot  ■  cos  d  —  cos  cot  ■  sin  d,  we  have 

ei  =  MI  cos  d  ■  sin^  cot  —  MI  sin  d  ■  sin  cot  -  cos  cot. 
Hence  by  equation  (3  3) 

Average  ei  =  MI  cos  d  av.  sin  "^cot  —  MI  sin  d  av.  sin  cot  cos  cot. 
The  average  value  of  sin  cot  cos  cot  is  zero  since  it  passes 
through  positive  and  negative  values  alike.  The  average  value 
of  sin  "^cot  is  ^ .     Therefore, 

4 


50         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

MI 

Average  ei  =  Power  = cos  d.  (38) 

It  is  more  convenient  to  have  this  product  expressed  in  terms  of 

the  effective  values  of  the  current  and  e.  m.  f.      Hence  substitute 

for  J5  and  J  their  values  as  given  by  equations  (36)  and  (37)  we 

have 

Power  =  £1  cos  d.  (39) 

Remark :  The  factor  cos  d  which  depends  upon  the  inductance 
and  resistance  of  the  circuit  which  is  receiving  the  power  [See 
Art.  50]  is  called  'Ca^  power  factor  of  the  circuit. 


CHAPTER  V. 


FUNDAMENTAL    PROBLEMS     IN    ALTERNATING    CURRENTS. 

49.  Problem  III.* — To  determine  the  e.  m.  f.  required  to  main- 
tain a  harmonic  alternating  current  in  a  non-inductive  circuit. 

Let    t  =  I  sin  cot  (<») 

be  the  given  harmonic  current.  The  required  e.  m.  f.  e  is  used 
wholly  to  overcome  the  resistance  R  of  the  circuit  and  it  is, 
therefore,  equal  to  Rt  so  that 

e  =  RI  sin  cot 
This  e.  m.  f.  is  harmonic,  its  maximum 
value  is  RIsLiid  it  is  in  phase  with  the  current 
i.  Thus  the  line  I,  Fig.  49,  represents  the 
given  harmonic  alternating  current  and  the  line 
MI  represents  the  e.  m.  f.  required  to  main- 
tain the  given  current  in  a  non-inductive  circuit. 

50.  Problem  IV. — To  determine  the  e.  m.  f. 
required  to  maintain  a  harmonic  alternating 
current  in  a  circuit  of  resistance  R  and  induc- 
tance L. 

Let  i^  I  sin  cot 

be  the  given  harmonic  current.  The  required  e.  m.  f.  consists  of 
two  parts,  namely : 

(i)  The  part  used  to  overcome  the  resistance  of  the  circuit. 

This  part  is  at  each  instant,  equal  to  Ri ;  it  is  in  phase  with  i 
and  its  maximum  value  is  RI. 

(2)  The  part  used  to  make  the  current  increase  and   decrease, 

or  briefly  to  overcome  the  inductance.     This  part  is,  at  each  in- 

di 
stant,  equal  to  Z  -^  according  to  equation  (3);  it  is  90°  ahead  of 

*  Problems  I.  and  II.  are  given  in  Chapter  I. 

(51) 


Fig.  49. 


52 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


i  in  phase  (see  Art.  44),  and  its  maximum  value  is  (oLI.     Let 

the  given  harmonic  alternating  current  i  be  represented  by  the 

line  I,  Fig  50.     Then  RI  is  the  line  which  represents  Ri;  oiLI 

di 
is  the  line  which  represents  L—j-;  and  the  line  'H  represents  the 

total  e.  m.  f.  required  to  maintain  the  given  current.     From  the 
diagram  we  have 

n  =  I^R^-^oi^n-  (40) 

in  which  ^  is  the  maximum  value  of  the  required  e.  m.  f.;  and 
further 

oiL 


tan^  = 


(41) 


in  which  Q  is  the  phase  difference  between  the  e.  m.  f.  and  current. 

The  effective  value  of  the  e.  m.  f.  is  £"=  -7=-,  and  the  effective 

V  2 

value  of  the  current  is  /=  — ^  [by   equations   (36)   and   (37)] 

■V  2 

therefore  substituting  V 2E  for  ^  and  \/ 2 /  for  I  in  equation 
(40)  we  have 

E=I>/R^  +  oj^D  (42) 

When  (oL  is  very  small  com- 
pared with  R  the  effect  of  induc- 
tance is  negligible  and  this  prob- 
lem IV.  reduces  to  the  problem 
III.  When  coL  is  very  large 
compared  with  R  the  angle  d 
approaches  90°  and  the  power 
EI  cos  6  becomes  very  small, 
although  E  and  /  may  both  be  considerable.  In  this  case  the 
current,  lagging  as  it  does  90°  behind  the  e.  m.  f.,  is  called  a 
wattless  current.  Thus  the  alternating  current  in  a  coil  of  wire 
wound  on  a  laminated  iron  core  is  approximately  wattl^ess. 

Corollary :    The  current  which  is  maintained  in  an  inductive 


Tig.  50. 


FUNDAMENTAL   PROBLEMS.  53 

circuit  by  a  given  harmonic  alternating  e.  m.  f.  is  a  current  of 

E 

which  the  effective  value  is  =-  by  equation  (42)  and 

which  lags  behind  the  e.  m.  f.,  by  the  angle  of  which  the  tangent 
is  -77   by  equation  (41). 

Remark:   The  relation  between  maximum  values  of  e.  m.  f. 

and  current  (harmonic)  is  in  every  case  the  same  as  the  relation 

between  effective  values,  and  henceforth  effective  values  will,  as  a 

rule,  be  used  in  equations  and  diagrams.     Maximum  values  will 

be  indicated  in  the  text  by  bold-faced  letters,  J5,  I,  N,  Q,  etc.; 

effective  values  by  the  letters  £,  I,  etc.,  and  instantaneous  values 

by  e,  i,  n,  g,  etc. 

51.  Problem  V. — To  determine  the  current  in  an  inductive  circuit  immediately 
after  an  harmonic  e.  m.  f.,  J^  sin  ut,  is  connected  to  the  circuit. 
The  current  which  can  be  maintained  by  tlie  given  e.  m.  f.  is 

i'  =  ^  sin  («/—  0)  (a) 

according  to  problem  IV.;  and  the  decaying  current 

i^^=^Ce  L  (lo)bis 

can  exist  in  the  circuit  independently  of  all  outside  e.  m.  f. ,  C  being  a  constant,  as 
shown  by  Problem  I.,  Chapter  I.  Therefore  the  current  which  can  exist  in  an  indue-, 
tive  circuit  upon  which  an  harmonic  e.  m.  f.  acts  i  =  i'  -|-  i"  or 

^  sin(6;if  — 0)  +  C£'    ^  (43) 


l/7?2  +  w2Z2 


in  which  6  is  the  Napierian  base,  0  is  the  angle  defined  by  equation  (41)  and  C  is  a 
constant.  This  constant  C  is  determined  by  the  condition  that  i  is  equal  to  zero  at 
the  instant  when  the  e.  m.  f.  is  connected  to  the  circuit.  Let  t'  be  the  given  instant 
at  which  the  harmonic  e.  m.  f.  begins  to  act  upon  the  circuit.     Substitute  the  pair  of 

values   -^  .~      >  in  equation  (43)  and  solve  for  C,  the  only  unknown  quantity  ;  then 

substituting  this  value  of  C  in  equation  (43)  we  have  the  expression  for  the  actual 
current  which  flows  in  the  circuit  during  the  time  that  the  maintained  cuiTent  is  being 
established.  In  a  very  short  time  after  the  e.  m.  f.  is  connected  to  the  circuit  the 
second  term  of  equation  (43)  disappears  and  the  value  of  the  cui'rent  at  each  instant 
is  given  by  the  first  term,  which  expresses  the  current  which  the  given  e.  m.  f.  can 
maintain. 


54 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


xnam 


52.  Problem  VI. — A  coil  of  resistance  R  and  inductance  L,  and 
a  condenser  of  capacity  y  are  connected  in  series  across  alternat- 
ing current  mains  as  shown  in  Fig.  51.     An  alternating  current 
flows  back  and  forth  through  the  coil  and  charges  the  conden- 
ser in    one    direction    and  the   other 
alternately.     The  problem  of  finding 
the    relation   between  the  current  in 
the  coil  and  the  e.  m.  f  between  the 
mains  is  reduced  to  its  simplest  form 
as  follows  : 

To  determine  the  e.  m.  f.  necessary 
to  make  the  charge  q  on  the  condenser 
vary  so  that 

q  =  Q  sin  (ot  (a) 

in  which  t  is  elapsed  time,  cot  is  an  angle  increasing  at  a  constant 
rate,  and  Q  is  the  maximum  value  of  the  charge  in  the  conden- 
ser. This  varying  charge  may  be  represented  by  the  projection 
of  the  rotating  Hne  Q,  Fig.  52,     The  current  in  the  circuit  is  the 

rate  -f-,  at  which  the  charge  on  the  con- 
at 

denser  changes. 


jnain 
Fig  51. 


That  is 
dq 
"^ 

or  from  equation  (a)  we  have 
2  =  coQ  cos  cot. 


(c) 


Fig.  52. 


That  is,  the  current  is  90°  ahead  of  q  in  phase,  its  maximum 
value  is  coQ  and  it  is  represented  by  the  line  wQ,  Fig.  52.  Us- 
ing the  symbol  I  for  the  maximum  value  of  the  current  we  have 

I=oyQ.  (d) 

The  required  e.  m.  f  is  at  each  instant  used  in  part  to  over- 
come the  resistance  R  of  the  coil,  in  part  to  cause  the  current  to 
increase  and  decrease  in  the  coil,  and  in  part  to  hold  the  charge 
on  the  condenser. 


FUNDAMENTAL   PROBLEMS. 


55 


1.  The  first  part  is  equal  to  Ri  at  each  instant.     It  is  in  phase 
with  i,  and  its  maximum  value  is  RI. 

2.  The  second  part  is  equal  to  Z  —  at  each  instant.     It  is  90° 
ahead  of  the  current  and  its  maximum  value  is  coLI. 

3.  The  third  part  is  equal  to  -x.  *   at  each    instant.     It  is  in 

phase  with  q  or  90°  behind  the  current,  and  its  maximum  value 

IS  ^  or  — r  irom  fd). 

Let  the  hne  I,  Fig.  53,  repre- 
sent the  current.  Then  RI  rep- 
resents the  portion  of  the  e.  m. 
f.  used  to  overcome  resistance, 
the  line  coLI  90°  ahead  of  the 
current  represents  the  e.  m.  f.  re- 
quired to  overcome  inductance 

and  the  line  — v  qo°  behind  the 

current  represents  the  e.  m.  f.  re- 
quired to  hold  the  charge  on  the 
condenser.  The  line  E  which  is 
the  vector  sum  of  RI,  (dLI  and 


Fig.  53. 


coj 


represents  the  total  required  e.  m.  f.     From  the  right  trian- 


ele  of  which  the  sides  are  RI,  cdLI ^ ,  and  J^,  we  have 


or 


and 


E=lil?+{.L-^^ 


coL  — 


tan  d  = 


~coJ 


(44) 
(45) 


R 


*  Since  q  =Je  according  to  equation  (17),  Chapter  I. 


56         THE  ELEMENTS   OF  ALTERNATING   CURRENTS. 

Corollary. — A  given  harmonic  e,  m,  f.  acting  upon  a  circuit 
containing  a  condenser  of  capacity  J,  inductance  L  and  resistance 
R  maintains  a  current  of  which  the  effective  value  is 

(46) 


J^^+H-^) 


and  which  lags  behind  the  e.  m.  f.  by  the  angle  Q  defined  by 
equation  (45).     The  quantity  coL  — —j  may  be  either  positive  or 

neg-ative  according-  as  cdL  or  — ^  is  the  greater  so  that  the  current 

may  be  either  behind  or  ahead  of  the  e.  m.  f.  in  phase.  In  fact 
the  Hmiting  values  of  d  are  d=  90°. 

The  oscillatory  current. — If  coL 7  =  0  then  the  impressed 

e.  m.  f  has  only  to  overcom.e  the  resistance  of  the  circuit  and 
problem  VI.  reduces  in  form  to  problem  III.  If  the  resistance  of 
the  circuit  in  this  case  were  negligibly  small  then  no  e.  m.  f.  at 
all  would  be  required  to  maintain  the  given  harmonic  current. 
Such  a  self  sustained  harmonic  current  is  called  an  oscillatory  cur- 
rent.    In  this  case  from  oiL  —  — ^=  o,  we  have : 

-=>I5  (47) 

or  since  a  =  27zf  [equation  (28)]  we  have 

This  equation  expresses  what  is  called  the  proper  frequejicy  of 
oscillation  of  the  inductive  circuit  of  a  condenser.  In  case  the 
resistance  of  the  circuit  is  not  zero,  which  is  of  course  the  only 
real  case,  then  the  only  current  which  can  exist  in  the  circuit  in- 
dependently of  any  impressed  e.  m.  f  is  a  decaying  oscillatory 
■cun^cnt  the  discussion  of  which  is  beyond  the  scope  of  this  text. 


FUNDAMENTAL   PROBLEMS. 


57 


The  character  of  this  decaying  oscillatory  curre.nt  is  shown  by 
the  curve,  Fig.  54.  The  ordinates  of  this  curve  represent  the 
successive  values  of  the  current  produced  when  a  charged  con- 
denser is  discharged  through  an  inductive  circuit. 


Fig.  54. 


53.  Electric  resonance. — By  inspecting  equation  (46)  we  see 
that  an  e.  m.  f  of  given  effective  value  E  produces  the  greatest 
current  in  the  circuit,  Fig.  5 1  {RL  and  J  given),  when  the  fre- 
quency is  such  that  mL is   zero.      This  production   of  a 

greatest  current  by  a  given  e.  m.  f.  at  a  critical  frequency  is  called 


110 

100 

Sso 

o_ 
^  60 

40 
20 

/ 

\ 

y 

V 

0 

l( 

)            c 

0         ,3 

0          4 

0          5 

0          6 

0         7 

3          i. 

0         9 

0          IC 

0         II 

0> 

Frec|uency 

Fig.  55. 

electrical  resonance.     Thus  the  ordinates  of  the  curve,  Fig.  55, 
represent  the  values  of  effective  current  at  various  frequencies 


58  THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

(abscissas).  The  curve  is  based  on  the  values  E=  200  volts, 
R=2  ohms,  ^  =  .352  henry  and  y=  20  microfarads.  The 
maximum  point  of  the  curve  is  not  a  cusp  as  would  appear  from 
the  figure,  but  the  maximum  is  so  sharply  defined  that  it  cannot 
be  properly  represented  in  so  small  a  figure.  When  the  frequency 
of  the  e.  m.  f.  is  zero,  which  is  the  case  when  a  continuous  e.  m. 
f.  acts  on  the  circuit,  the  current  is  zero  except  for  the  very  slight 
current  which  is  conducted  through  the  dielectric  between  the 
condenser  plates.  When  the  frequency  of  the  e.  m.  f  is  very 
great  the  current  approaches  zero,  inasmuch  as  a  very  small  cur- 
rent of  high  frequency  must  increase  and  decrease  at  a  very  rapid 
rate  and  to  produce  this  rapid  increase  and  decrease  a  very  great 
e.  m.  f.  is  required.  At  low  frequency  the  current  is  kept  down 
in  value  by  the  condenser  and  at  high  frequency  the  current  is 
kept  down  in  value  by  the  inductance. 

An  important  point  in  connection  with  electrical  resonance  is 

that  the  e.  m.  f ,  — :    at  the  condenser  terminals  may  be  greatly 

in  excess  of  the  e.  m.  f.  E,  which  is  maintaining  the  current. 

Thus  the  line — ^,  Fig.  53,  may  be  considerably  longer  than  the 

line  E. 

Example :  A  coil  of  .352  henry  inductance  and  2  ohms  resist- 
ance, and  a  condenser  of  20  microfarads  capacity,  are  connected 
in  series  between  alternating  current  mains.  The  critical  fre- 
quency of  this  circuit  is  60  cycles  per  second,  according  to  equa- 
tion (48).  The  e.  m.  f  between  the  mains  is  200  volts  and  its 
frequency  is  60  cycles  per  second.  The  current  in  the  circuit  is 
100  amperes  according  to  equation  (46)  and  the  effective  e.  m.  f. 

between  the  condenser  terminals  is  13,270  volts  (  =  — -I. 

Remark :  At  critical  frequency  coL  —  — -  =  o  and  equation 
(46)  becomes  simply  ^  =  -„• 


FUNDAMENTAL   PROBLEMS.  59 

Remark:  While  an  e.  m.  f.  is  being  established  between  the 
plates  of  a  condenser  the  dielectric  is  subjected  to  an  increasing- 
electrical  stress  and  this  increasing  electrical  stress  is  exactly 
equivalent,  in  its  magnetic  action,  to  an  electric  current  flowing 
through  the  dielectric  from  plate  to  plate.  Magnetically,  there- 
fore, a  circuit  containing  a  condenser  is  a  complete  circuit.  In- 
creasing (or  decreasing)  electrical  stress  is  called  displacement 
current. 

Mechanical  resonance. — The  mechanical  analogue  of  a  con- 
denser and  an  inductance  connected  in  series  is  described  in  Art. 
1 8.  The  body  fixed  to  the  end  of  a  clamped  flat  spring  (see 
Fig.  a,  Art.  1 8)  will  vibrate  at  a  definite  frequency  when  pulled 
to  one  side  and  released.  This  frequency  is  called  the  proper 
frequency  of  vibration  of  the  body. 

If  a  periodic  force  of  given  maximum  value  and  given  fre- 
quency acts  upon  the  body  (Fig.  a,  Art.  i8)  the  body  will  be  set 
vibrating  at  the  same  frequency  as  that  of  the  force,  and  the 
violence  of  the  motion  will  be  greatest,  for  the  given  value  of  the 
periodic  force,  when  the  frequency  of  the  force  is  equal  to  the 
proper  frequency  of  the  body.  Under  these  circumstances  the 
bending  force  (periodic)  acting  upon  the  spring  may  greatly  ex- 
ceed in  value  the  outside  force  which  keeps  the  body  in  motion. 
Thus,  if  a  narrow  strip  of  window  glass  is  clamped  at  one  end, 
loaded  at  the  other  end,  and  set  vibrating  by  shght  pushes  of  the 
finger,  the  strip  is  quickly  broken  if  the  frequency  of  the  pushes 
is  the  same  as  the  proper  frequency  of  oscillation  of  the  loaded 
strip.  The  breaking  of  the  glass  shows  that  the  bending  force 
acting  upon  it  reaches  values  greatly  in  excess  of  the  mere  push 
of  the  finger. 

Problems. 

I.  A  circuit  has  inductance  Z=o.2  henry  and  a  resistance 
R^6  ohms.  Calculate  the  current  produced  by  lOO  volts,  the 
frequency  being  6o  cycles  per  second.     Calculate  the  phase  dif- 


6o         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

ference  between  the  e.  m.  f.  and  current.      Calculate  power  de- 
veloped. 

2.  A  circuit  has  i6o  ohms  resistance  and  .2  henry  inductance. 
Calculate  the  power  factor  of  the  circuit  for  a  frequency  of  60 
cycles  per  second. 

3.  An  e,  m.  f.  of  20,000  volts  acts  on  a  receiving  circuit  q>{ 
which  the  power  factor  is  .85.  Find  the  component  of  e.  m.  f. 
parallel  to  the  current  and  the  component  of  e.  m.  f.  perpendicu- 
lar to  the  current. 

4.  A  i6-candle  lamp  of  which  the  resistance  is  50  ohms  re- 
quires I  ampere  of  current.  This  lamp  is  to  be  connected  direct 
to  1,000-volt  mains  (frequency  133)  in  series  with  a  condenser. 
Calculate  the  capacity  of  the  condenser  in  order  that  the  required 
current  of  one  ampere  may  flow.  Calculate  the  current  when 
two  lamps  are  connected  in  series  with  this  condenser ;  when 
three  lamps  are  connected  in-  series  with  the  condenser ;  and 
when  the  condenser  is  connected  direct  to  the  mains  without  any 
lamps  in  circuit. 

5.  An  inductance  of  3  henrys  is  connected  between  1,000-volt 
mains  (frequency  60),  calculate  the  current  in  the  circuit  when  the 
resistance  is  zero  (negligibly  small),  when  the  resistance  is  20 
ohms,  and  when  the  resistance  is  50  ohms. 

6.  Calculate  the  power  taken  from  the  mains  in  each  case 
given  in  problems  4  and  5. 

7.  Two  condensers  of  0.5  and  0.05  microfarads  capacity  re- 
spectively are  connected  in  series  between  1,100-volt  mains. 
Find  the  effective  e.  m.  f.  between  the  terminals  of  each  con- 
denser. 

8.  A  quadrant  electrometer  of  which  the  capacity  is  negligible 
is  connected  to  the  terminals  of  a  o.  56-microfarad  condenser. 
This  condenser  is  connected  between  mains  in  series  with  a 
o.o6-microfarad  condenser.  The  quadrant  electrometer  indicates 
107  volts.     What  is  the  e.  m.  f.  between  the  mains  ? 

9.  A  quadrant  electrometer  has  an  electrostatic  capacity  of 


FUNDAMENTAL  PROBLEMS.  6 1 

.00006  microfarads.  It  is  connected  to  an  alternator  through  a 
wire  of  50,000  ohms  non-inductive  resistance.  Find  the  percent- 
age error  of  the  e.  m.  f.  indications  of  the  electrometer  due  to 
loss  of  e.  m.  f  in  the  resistance  the  frequency  being  60  cycles  per 
second. 

10.  The  above  quadrant  electrometer  is  connected  to  the  alter- 
nator (60  cycles  per  second)  through  a  coil  of  1.5  henrys  induc- 
tance and  of  negligible  resistance.  Find  the  percentage  error  of 
the  indications  of  the  electrometer  due  to  loss  of  e.  m.  f  in  the 
inductance. 

11.  An  electrodynamometer  has  an  inductance  of  .0001 
henry ;  it  is  to  be  used  as  a  voltmeter  at  a  frequency  of  60  cycles 
per  second.  What  non-inductive  resistance  must  be  connected 
in  series  with  the  instrument  to  reduce  the  inductance  error  of  its 
readings  to  yqJo. 


CHAPTER  VL 

THE   USE   OF   COMPLEX   QUANTITY.    STEINMETZ'S  METHOD. 

54.  Methods  in  alternating  currents.  The  graphical  and  the 
trigonometrical  method. — All  fundamental  problems*  in  alternat- 
ing currents  may  be  solved  by  the  graphical  method  in  which  the 
various  e.  m.  f.'s,  currents,  etc.,  are  represented  by  lines  in  a 
diagram  and  the  required  results  are  measured  off  as  in  graphical 
statics.  In  practical  problems,  however,  the  different  quantities 
under  consideration  differ  so  greatly  in  magnitude  that  it  is  diffi- 
cult to  scale  off  results  with  any  degree  of  accuracy.  The 
graphical  method  is,  however,  particularly  useful  for  giving  clear 
representations,  and  trigonometric  formulas  may  be  used  in  con- 
nection with  graphical  diagrams  in  every  case. 

The  trigonometric  formuhe  in  the  more  complicated  problems, 
however,  become  very  unwieldy  and  are  not  suitable  for  easily 
obtaining  numerical  results. 

Steinmetz' s  method. — Numerical  results  in  alternating  current 
calculations  are  most  easily  obtained  by  means  of  a  method  de- 
veloped mainly  by  Steinmetz  in  which  complex  quantity  is  used. 
This  method  is  purely  algebraic  and  is  called  by  Steinmetz  the 
symbolic  method. 

55.  Simple  quantity.  Complex  quantity, — A  simple  quantity 
is  a  quantity  which  depends  upon  a  single  numerical  specification. 
Simple  quantities  are  often  called  scalar  quantities.  A  complex 
quantity  requires  two  or  more  independent  numerical  specifica- 

*  The  fundamental  problems  are  those  which  treat  of  harmonic  e.  m.  f.  and  har- 
monic current.  It  is  a  mistake  to  suppose  that  differential  equations  furnish  a  method 
for  treating  alternating  currents  distinct  from  the  three  methods  mentioned  above.  In 
the  application  of  differential  equations  the  first  step  is  to  derive  one  or  more  harmonic 
expressions  for  e.  m.  f.  and  current  and  the  subsequent  development  is  precisely  the 
one  or  the  other  of  the  above-mentioned  methods. 

(62) 


USE   OF   COMPLEX   QUANTITY.  63 

tions  to  entirely  fix  its  value.  For  example,  if  wealth  depends 
upon  the  possession  of  horses  (Ji)  and  cattle  (c),  then,  if  no 
agreement  exists  as  to  the  relative  value  of  horses  and  cattle  (in 
fact  any  such  agreement  is  essentially  arbitrary),  the  specification 
of  the  wealth  of  an  individual  would  require  the  specification  of 
both  horses  and  cattle.  Thus  the  wealth  of  an  individual  might 
be  ^h  -f  lOOi;.  The  two  or  more  numbers  which  go  to  make  up 
a  complex  quantity  are  called  the  elements  of  the  quantity. 

Addition  and  subtraction. — Two  complex  quantities  are  added 
or  subtracted  by  adding  or  subtracting  the  similar  numerical 
elements  of  the  quantities.  Thus  5/^  +  looc  added  to  6h  +  i  <,c 
gives  iih  -{-  ii^c  and  2h  +  25^  subtracted  from  ^h  +  lOOi::  gives 

Midtiplication  and  division. — Consider  two  complex  quantities 
^h  +  2c  and  3/^  +  4^  in  which  h  and  c  are  independent  incommen- 
surate units,  say  horses  and  cattle.  Multiplying  the  first  of  these 
expressions  by  the  latter,  using  the  ordinary  rides  of  algebra,  we 
have  : 

{^h  +  2c)  {T,h  -\-  4c)  =  1 5/^^  +  6/ic  +  20ck  -t-  8c^. 

Now  in  general  the  squares  and  products  of  units  c^,  /i^,  ch  and 
Jic  have  no  meaning ;  so  that  the  significance  of  the  product  of 
two  complex  quantities  depends  upon  arbitrarily  cJiosen  meanings 
for  these  squares  and  products  of  nnits. 

56.  Vectors. — A  vector  is  a  quantity  which  has  both  magni- 
tude and  direction.  A  vector  may  be  specified  by  giving  its  com- 
ponents in  the  direction  of  arbitrarily  chosen  axes  of  reference. 
In  specifying  a  vector  by  its  components,  it  is  necessary  to  have 
it  distinctly  stated  which  is  its  x  component  and  which  is  its  y 
component.*  This  may  be  done  either  by  verbal  statement  or 
by  marking  one  of  the  components  by  a  distinguishing  index. 
Further  it  is  allowable  to  connect  the  two  components  with  the 
sign  of  addition.  Thus  a  -\-jb  is  a  specification  of  the  vector 
of  which  the  x  component  is  a  and  the  y  component  is  b ;  the 

*We  are  at  present  concerned  only  with  vectors  in  one  plane. 


b 


O 


64         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

index  letter  /  being  used  to  mark  the  y  component.  This  ex- 
pression of  a  vector  is  a  complex  quantity  the  independent  units 
of  which  are  vertical  and  horizontal    distances  or  components. 

For  example,  the  vector  OP,  Fig.  56, 
is  specified  by  the   horizontal  com- 
ponent a  and  the  vertical  component 
'  b.     The    vector    may   therefore    be 

specified   by  the   expression    a  -\-jb 
a  ac      the  index  j  being  used  to  show  that 

b  is  the  vertical  component. 

Numerical  value  and  direction  of  a 

Fig.  56.  vector. — The   numerical   value    of  a 

vector  is  the  square  root  of  the  sum 

of  the  squares  of  its  components.     Thus  the  numerical  value  of 

the   vector  a  -\-jb  is  ^ a?"  -\-  b'^.         ' 

The  angle  between  the  vector  and  the  x  axis  is  the  angle,  d, 

of  which  the  tangent  is  -,  that  this  tan  d  ^  —.      This    matter    of 
^  a  a 

value  and  direction  of  a  vector  is  an  important  consideration  in 
alternating  current  problems. 

Addition  and  subtraction  of  vectoj's. — The  sum  of  a  number  of 
vectors  is  a  vector  of  which  the  x  component  is  the  sum  of  the 
X  components  of  the  several  vectors,  and  of  which  the  y  com- 
ponent is  the  sum  of  the  y  components  of  the  several  vectors. 
Thus  the  sum  of  the  vectors  a  +jb,  a'  -\-jb',  a"  +jb"  is 

{a  +  a'  +  a")  -^j{b  +  b'  +  b"). 

The  difference  of  the  two  vectors  a  -\-jb,  a'  -\-Jb'  is 

{a  -  a')  -^j{b  -  b'). 

Multiplication  of  vectors. — Consider  the  two  vectors  a  -\-jb  and 
a'  -\-jb'.  Multiply  these  two  expressions,  using  the  formal  rules 
of  algebra,  and  we  have  for  the  product 

aa'  -\-jab'  -\-ja'b  -\-j^bb'. 

Each  term  in  this  product  must  be  interpreted  arbitrarily  or 


USE  OF  COMPLEX  QUANTITY.  6$ 

according  to  convention  in  order  that  this  product  may  have  a 
definite  meaning. 

In  the  first  place,  we  may  take  aa' ,  which  is  not  affected  by 
the  index/,  to  be  a  horizontal  quantity  and  we  may  take  ad'  and 
a'd  to  be  vertical  quantities  or  vertical  components  of  a  vector. 
As  to  the  term  j'^do'  we  may  note  that  the  index  letter  j,  used 
once,  indicates  that  a  quantity  is  vertical,  while  without  the  index 
/  the  quantity  would  be  understood  to  be  horizontal  and  to  the 
right  (see  Fig.  56).  That  is,  the  letter  /  may  be  thought  of  as 
turning  a  quantity  through  90°  in  the  positive  direction,  that  is, 
counter  clockwise.  It  is,  therefore,  convenient  to  think  of  the 
letter/  when  used  twice  {jjdd'  or  j'^bb'')  as  turning  the  quantity 
upon  which  it  operates  through  1 80°  or  as  reversing  its  direction 
and,  therefore,  its  algebraic  sign.  That  is,  j^bb'  is  to  be  taken  as 
equal  to  —  bb'  or 

so  that  the  product  of  the  two  vectors  a  +ja'  and  b  -\-jb'  is  to 
be  interpreted  as  the  vector 

{ab-a'b')^j{ab'  +  a'b). 

Quotient  of  two  vectors. — Consider  the  quotient 

a  -\-jb 
a'  -\-jb' 

multiply  both  numerator  and  denominator  by  a'  —jb',  remem- 
bering that  j^  =  —  I,  and  we  have 

a  -\-jb       aa'  +  bb'        .  la'b  —  ab'\ 
a'  ^jb'  ^   a!^  +  ^'2   +-^  \  fl'2  _^  jjn  ) 

which  leads  to  the  conception  of  the  quotient  of  two  vectors  as 
a  vector  of  which  the  x  component  is 


and  the  J  component  is 


aa' 

+  bb' 

a" 

+  b'^ 

a'b- 

-ab' 

a"  ^ 


66 


THE   ELEMENTS   OF   ALTERNATING  CURRENTS. 


We  shall  now  apply  the  symbolic  method  to  the  discussion  of 
Problems  IV.  and  VI.  in  order  to  illustrate  its  use  and  to  point 
out  a  few  definitions.  In  the  next  chapter  this  method  will  be 
applied  to  more  complicated  problems. 

57.  Application  of  the  symbolic  method.  Problem  IV.  again. — 
Consider  an  alternating  current  /  (effective  value),  in  a  curcuit  of 
which  the  resistance  is  R  and  the  inductance  is  L.  Let  the  line 
/,  Fig.  57,  which  represents  the  current,  be  chosen  as  the  .sr-axis 
of  reference..  Let  E  be  the  e.  m.  f  which  is  maintaining  the  cur- 
rent. The  .^-component  of  E  is  RI  and  the  jj/-component  of  E 
is  (iiLL     Therefore  : 

E={R^jo>Ly.  (49) 

Problem  VI.  again. — In  this  case,  as  shown  in  Fig.  58,  the 
;r-component  of  E  is  RI  and  the  jj/-component  of  E  is  (oLI y 

Therefore :  E=\R^j{mL-  A)]  /•  (50) 

Impedance,   resistance,  reactajice. — The   complex  quantity  by 
which  /  is  multiplied  to  give  E 
is  called  the  impedance  of  a  cir- 
cuit.    Thus    the    impedance    of 
the  circuit  discussed  in  problem 


Fig.  57. 


Fig.  58. 


IV.  is  R  -{-jcoL,  and  the  impedance  of  the  circuit  discussed  in 
problem  VI.  \s  R  -\-j  y  coL ^  I. 

The  quantity  by  which  the  current  is  multiplied  to  give  the 
component  of  E  parallel  to  the  current  is  called  the  resistance  of 


USE  OF  COMPLEX  QUANTITY.  67 

the  circuit.  Thus  the  resistance  of  the  circuits  discussed  in 
Problems  IV.  and  VI.  is  R. 

The  quantity  by  which  the  current  is  multipHed  to  give  the 
component  of  E  perpendicular  to  the  current  is  called  the  reac- 
tance of  the  circuit.  Thus  the  reactance  of  the  circuit  discussed 
in  problem  IV.  is  coL  and  the  reactance  of  the  circuit  discussed 

in  problem  VI.  is  o)L -^.     The  reactance  due  to  a  condenser 

coj 

is  negative.  Reactance  is  frequently  represented  by  the  letter  x. 
It  is  to  be  kept  in  mind  that  reactance  depends  upon  the  fre- 
quency /  I  =  — :;  I  as  well  as  upon  inductance  and  capacity. 

Ad?nittance.  Conductance.  Susceptance.  Consider  a  circuit  of  resistance  r  and 
reactance  x  in  which  an  alternating  current  /  is  maintained  by  an  e.  m.  f.  E.     Then 

I  _ 


/= 


>+yx 


'=( 


^2  -j-  x2         -^^2-1- 


(51) 


The  quantity,    ^      — ^ ,  by  which  E  is  multiplied  to  give  the  component  of  /paral- 
lel to  E  is  called  the  conductance  of  the  circuit. 

X 

The   quantity,  ^^— ; ^  ,  by  which  E  is  multiplied  to  give  the  component  of  /per- 
pendicular to  E  is  called  the  susceptance  of  the  circuit. 
Equation  (51)  is  sometimes  written 

(52) 

in  which  S=^2j.    ^2  (53) 


and  .  ■^  =  1^J^2  (54) 

The  complex  quantity^ — jb  is  called  the  admittance  of  the  circuit.  * 

Remark :  Resistance  and  reactance  are  both  expressed  in 
ohms,  and  the  numerical  value  of  impedance,  which  is  the  square 
root  of  the  sum  of  the  squares  of  resistance  and  reactance,  is' 


68         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

expressed  in  ohms.  Thus  a  circuit  of  which  the  inductance  is 
.02  henry,  has  a  reactance  of  7.54  ohms  at  a  frequency  of  60 
cycles  per  second  ;  a  condenser  of  which  the  capacity  is  2  mi- 
crofarads has  a  negative  reactance  of  1325  ohms  at  a  frequency 
of  60  cycles  per  second. 

Problems. 

1 .  Separate  the  components  of  the  complex  expression 

a'+jd' 

2.  A  circuit  carrying  current  at  a  given  frequency  has  a  resis- 
tance of  5  ohms  and  a  reactance  of  10  ohms.  Ten  amperes  of 
current  are  flowing.  Calculate  the  components  of  the  e.  m.  f. 
and  calculate  the  full  effective  value  of  the  e.  m.  f 

3.  At  what  frequency  is  the  reactance  due  to  .352  henry  in- 
ductance equal  (but,  of  course,  opposite  in  sign)  to  the  reactance 
of  a  20-microfarad  condenser? 

4.  Separate  the  components  of  the  complex  expression 


CHAPTER   VII. 

FURTHER   FUNDAMENTAL   PROBLEMS. 

58.  Problem  VII.  Coils  in  series.  Electromotive  force  coU' 
sumed  in  transmission  lines. — An  e.  m.  f.  E  acts  upon  two  coils 
in  series  as  shown  in  Fig.  59.     For  example,  one  coil  may  rep- 


FiG.  59. 

resent  a  transmission  line  and  the  other  a  receiving  circuit ;  the 
total  resistance  and  reactance  of  the  two  transmission  lines  being 
looked  upon  as  located  in  one,  only,  of  the  coils  for  the  sake  of 
simplicity.  It  is  required  to  find  E^  and  E^  each  in  terms  of  E, 
Tj,  r^,  x^  and  x^.  Let  /  be  the  current  in  the  circuit.  The  gen- 
eral relation  between  E,  E^,  E.^  and  /  is  shown  in  Fig.  60.  The 
symbolic  method,  however,  affords  the  easiest  and  simplest  solu- 
tion of  the  problem  as  follows  : 

E=E^  +  E^.  (a) 

This  is  a  vector  equation  and  expresses  the  fact  that  E  is  the  re- 
sultant of  E^  and  E^  as  shown  in  Fig.  60 


^2  =  (^^2  +  A)  ^• 
(69) 


(b) 


70         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

These  equations  (b)  and  (c)  come  from  problem  IV,,  Chapter  VI. 
From  equations  (a),  (b)  and  (c)  we  have 

E 
1= 7 ^  .  Cd) 

^1   +  ^^2  +-/(-^l  +'^2) 

Substituting  this  value  of  /  in  (b)  and  (c)  we  have 

E  =  E ^^  +-^'-^^ (€) 

'        ^1  +  ^2  +-/(-^i  +  ^2) 
and 

E  =  E ^^'  '^'^^' (f) 

'         ^\  +  ^^2  +-/(-^i  +  -^'2)  ^ 

These  equations  (e)  and  (f)  express  the  e.  m.  f.'s  £^  and  E^  in 

terms  of  the  known  quantities  E,  r^,  7\,  x\  and  x^.     For  purposes 

of  numerical  calculation  (e)  [and  likewise  (f )]  must  be  separated 

into  components,  that  is,  into  real  and  imaginary  parts,  and  the 

numerical  value  of  E-^  (and  likewise  of  E^)  is  then  found  by  taking 

the  square  root  of  the  sum  of  the  squares  of  these  components. 

Thus,  multiplying  numerator  and  denominator  of  (e)  by  r^  +  r^ 

— JX^i  ~^  ^'2)  ^^  remove  j  from  the  denominator  and  may  then 

separate  the  components,  namely 

'~        (r,  +  r,f  +  (X,  +  .r,)^    +-^^  "(r,  +  r,)^  +  (^,  +  ;.,)^-  ^^^ 

The  first  term  of  this  expression  is  the  component  of  E^  par- 
allel to  E  and  the  second  term,  dropping  j,  is  the  component  of 
Ej^  perpendicular  to  E  and  the  numerical  value  of  E^^  is  the  square 
root  of  the  sum  of  the  squares  of  these  components.*  The  final 
result  is  so  complicated  that  it  is  of  no  use  whatever  in  giving  a 
conception  of  the  phenomenon  under  consideration.  It  is  useful 
only  when  it  is  desired  to  carry  out  numerical  calculations. 
It  is,  indeed,  generally  the  case  that  the  use  of  the  symbolic 
method  in  the  solution  of  alternating  current  problems  is  simple 
and  instructive  in  its  initial  steps  only,  while  the  final   solution 


^£,  (numencal  value)  =  l^^J^jL_iA_L^   ^., 


FURTHER   FUNDAMENTAL   PROBLEMS. 


71 


itself  is  unintelligible.  The  final  results  will,  therefore,  be  written 
out  in  full  only  when  the  student  is  expected  to  use  them  in 
numerical  calculations. 

The  following  simple  cases  of  the  problem  under  consideration 
are  particularly  interesting. 

1 .  When  x^  =  o  and  x^__  =  o  ;  then  E,  E^,  E^,  and  /  are  parallel 
and  E^  E^-\-  E^  (numerically). 

X  X 

2.  When  -^  =  —  ;  then  E,  E^,  and  E^  are  parallel  to  each 
other  and  all  are  ahead  of  /  in  phase,  by  the  angle  of  which  the 

X 

tangent  is  — \     In  this  case,  also,  E^=  E^-\-  E^  (numerically). 


3.  When  —  is  very  small  and  -j  very  large  ;  then  E^  is  parallel 

to  /  and  it,  is  at  right  angles  to  /,  as  shown  in  Fig.  6 1 .  This  figure 
is,  of  course,  a  particular  case  of  Fig.  60.  In  the  present  case  the 
numerical  relation  between  E,  E^  and  E^'vs,  E  =  V  E^  +  E^  ;  and 
when  E^  is  small  E^  is  sensibly  equal  to  E  numerically. 

Example :  A  transmission  line  of  large  resistance  r^  and  small 
reactance  x^  supplies  current  to  a  receiving  circuit  of  large  reac- 

X  X 

tance  x^  and  small  resistance  7^^,  so  that  — ^  is  very  small  and  —^  is 

very  large.  The  e.  m.  f  E^,  Fig.  61,  consumed  in  the  Hne  is  due 
almost  wholly  to  resistance  and  if  E^  is  not  very 
large  then  E^  is  very  nearly  equal  to  E.  That  is 
the  resistance  drop  in  a  transmission  produces  but 


Fig.  61. 


Fig.  62. 


very  little  diminution  of  e.  m.  f.  at  the  terminals  of  a  receiving 
circuit  of  lars^e  reactance. 


72         THE    ELEMENTS   OF   ALTERNATING   CURRENTS. 


X  ■■  —^~  X 

When  —^  is  very  large  and  ~  very  small  the  state  of  affairs 


is  shown  in  Fig.  62,  and  ^=  \/ E^  -\-  E^. 

Example :  A  transmission  line  of  large  reactance  x^  and  small 
resistance,  1\,  supplies  current  to  a  receiving  circuit  of  large  resis- 
tance r^  and  small  reactance  x^.  The  e.  m.  f.,  E-^^,  Fig.  62,  con- 
sumed in  the  line  is  due  almost  wholly  to  reactance  and  if  E^  is 
not  very  large  then  E^  is  very  nearly  equal  to  E.  That  is  the 
reactance  drop  in  a  transmission  line  affects  the  e.  m.  f  at  the  ter- 
minals of  a  lar^e  resistance  receiving  circuit  but  little. 


X,  .    .  .         .  .  .  x„ 


5.  When  —  is  large  and  positive  and  -^  is  large,  but  negative. 

Then  E^  is  nearly  90°  ahead  of  /  and  E^  is  nearly  90°  behind  /, 
as  shown  in  Fig.  63.     The  figure  shows  the  limiting  case  for 

X  X 

which  —  =-|-  00   and  —  = —  co  .     In  this  case  the  total  e.  m.  f. 

r  r 

'  1  '2 

E  is  numerically  equal  to  the  difference  between  E^  and  E^. 

Examples :   (a)  A  transmission  line 
of  small  resistance  r^  and  large  react- 
Bj  ance  x^  supplies  current  to  a  conden- 

ser.    The  state  of  affairs  is  shown  in 
J   '  ^  Fig.  63  and  the  e.  m.  f.  E^  at  the  ter- 
^  minals  of  the  condenser  exceeds  the 

generator  e.  m.  f  E  by  the  amount 

^     I  Ey     Therefore    reactance    drop   in    a 

Bif  transmission  line  increases  the  e.  m.  f. 

at  the  terminals  of  a  receiving  circuit 

_  of  negative  reactance. 

Fig.  63.  ^ 

{b)  A  transmission  line  of  small  re- 
sistance r^  and  large  reactance  x^  supplies  current  to  a  synchro- 
nous motor  running  at  light  load  ;  e.  m.  f.  of  motor  being  less 
than  e.  m.  f.  of  generator.  In  this  case  the  e.  m.  f  at  the  termi- 
nals of  the  receiving  circuit  is  nearly  90°  behind  the  current  in 
phase,  as  in  case  of  the  condenser,  and  the  e.  m.  f.  at  the  re- 


\1 


FURTHER   FUNDAMENTAL   PROBLEMS. 


73 


ceiving  circuit  terminals  is  increased  by  the  reactance  drop  in 
the  line. 

(c)  An  inductance  is  connected  in  series  with  a  condenser  be- 
tween alternating  current  mains.  If  the  resistance  of  the  circuit 
is  comparatively  small  the  e.  m.  f.  at  the  condenser  terminals  is 
nearly  equal  to  the  sum  of  the  e.  m.  f.  between  mains  plus  the  e.  m. 
f.  between  the  terminals  of  the  inductance.    (See  Arts.  52  and  53.) 

59.  Problem  VIII.  Coils  in  parallel. — A  given  alternating  cur- 
rent /  divides  between  two  circuits  in  parallel,  as  shown  in  Fig. 
64.  It  is  required  to  find  /j  and  L^  each  in  terms  of  /,  r^,  Tg,  ^,, 
and  .ar^.     Let  E  be  the  e.  m.  f  between  the  branch  points.     The 


Fig.  64. 


Fig.  65. 


general  relation  between  /,  7^,  /^,  and  B  is  shown  in  Fig.  65. 
The  symbolic  method,  however,  affords  the  easiest  and  simplest 
solution  of  the  problem  as  follows  : 


L  = 

1\  +JX. 


(a) 

(b) 
(c) 


whence 


+ 


^2  +J^2 


or 


E=I 


(d) 


74 


THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 


Substituting  this  value  of  E  in  (b)  and  (c)  we  have 
'  ^'i(^'i  +  ^'2)  -  -^i('^i  +  -^'2)  +  /  [^1(^1  +  -^'2)  +  -^1(^1  +  ^2)] 


(e) 


and  a  similar  expression  for  I^.  In  this  expression  for  I^  the 
index  letter  /  may  be  removed  from  the  denominator  by  multi- 
plying numerator  and  denominator  by 

'\{^x  +  ^'2)  -  ^iG'^'i  +  •^'2)  -/  Va^^^x  +  ^2)  +  ^i(^^i  +  ^2)] 
when  the  components  of  /^  parallel  to  /  and  perpendicular  to  / 
may  be  separated  exactly  as  in  problem  VII. 

The  following  simple  cases  of  the  problem  under  consideration 
are  interesting. 

I .  When  x^  =  o  and  ;f ^  =  o  ;    then  /,  /^  I^  and  B  are  parallel 
and  I  =  I^-\-  I^  (numerically). 
x^       x„ 
r,       r 


2.  When  —  =  —  then  /,  I^,  and  I^  are  parallel  to  each  other 


1 
and  all  behind  E  in  phase  by  the  angle  of  which  the  tangent  is 


— .      In  this  case,  also,  I  =  I^-\- 1^  (numerically). 


x„ 


3.  When  -j-  is  very  small  and  -f  is  very  large  or  vice  versa.     In 
^'1  ''2 

this  case  /^  is  parallel  to  E  and 

E     ^         /^  is  90°  behind  E  or  vice  versa 

as   shown   in  Figs.   66  and  67. 

These  figures  are  particular  cases 


Fig.  66. 


Fig.  67. 


of  Fig.  65.  In  the  present  case  the'  numerical  relation  between  /, 
/j,  and  /g  is  /  =  ^^ I ^  +  //  and  when  either  I^  or  I^  is  small  the 
other  is  sensibly  equal  (numerically)  to  /. 


FURTHER   FUNDAMENTAL   PROBLEMS. 


75 


X  X 

4.  When  —  is  large  and  positive  and  —  large  but  negative  then 
^1  ^2 


E 


Fig.  68. 


/j  is  90°  behind  E,  and  I^  is  90  degrees  ahead  of  E,  as  shown 
in  Fig.  68.     In  this  case  /  =  /g  —  I^ 
(numerically),  that  is,  the  sum  of  the 
currents   in  the  branches   exceeds   the 
total  current  /. 

Examples  of  case  ^ :  A  condenser 
and  an  inductance  are  connected  in 
parallel  in  an  alternating  current  cir- 
cuit. The  current  /  in  the  circuit 
divides  in  the  two  branches  formed  by 
the  inductance  and  the  condenser.  The 
currents  I^  and  I.^  in  these  two  branches  are  nearly  opposite  to 
each  other  in  phase  and  the  current  /  is  numerically  equal  to 
the  difference  of  /^  and  I^. 

If  the  frequency  of  the  alternating  current  /  is  such  that  coL 

and  — -J.  are  equal  then  /^  and  I^  will  be  nearly  equal ;  and  /^  and 

/g  will  each  be  very  much  greater  than  /. 

60.  Problem  IX.       Compensation   for  lagging  currents. — The 

current  I^,  Fig.  69,  in  a  receiving  circuit  lags  behind  the  e.  m.  f , 

E  which  acts  upon  the  ter- 
minals of  the  receiving  cir- 
cuit. If  an  auxiliary  circuit 
having  negligible  resistance 
and  negative  reactance  is  con- 
nected between  the  terminals 
of  the  receiving  circuit  {i.  e., 
in  parallel  with  the  receiving 

circuit),  then  this  auxiliary  circuit  will  take  the  current  /^  which 

is  90°  ahead  of  E,  and  the  total  current  given  by  the  alternator 

will  be  reduced  to  /  in  phase  with  E. 

Discussion  :  From  problem  IV.  we  have 


76 


THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 


I^  = 


E 


^2   +-7-^2 


or 


/ 


=4 


r 

2  ""i     172 


-J . 


x„ 


x„ 


2     +^2 

SO  that  the  component  of  /  perpendicular  to  E  is 


(a) 

(b) 


;ij; 


+   ^2 


Further,  if  the  auxiliary  circuit  has  no  perceptible  resistance  and 

.    E         . 
X,  is  its  reactance  then  the  numerical  value  of  /  is  —  and  in  or- 
^  ^      x^ 

der  that  /  may  be  parallel  to  E  we  have 


Ex„ 


+    ^2' 


E 

'  X, 


or 


X  =  — 


^^2'  +  ^2 
x„ 


which  expresses  the  value  of  the  reactance  (negative)  which  the 
auxiliary  circuit  must  have  to  compensate  for  the  lagging  cur- 
rent in  the  receiving  circuit. 

Example :  An  alternator  giving  2000  volts  at  a  frequency  of 
133  cycles  per  second  {co  =  27r  x  133)  furnishes  current  to  a  cir- 
cuit of  which  the  resistance  r^  is  100  ohms  and  the  inductance  is 
.03   henry.     The  reactance  (nL  of  the  circuit  is  therefore  27r  x 

133  X  .03  or  25  ohms.     The  value  of  ;r^  I  =  — ^1    required    to 

compensate  for  lagging  cur- 
rent in  this  circuit  is  425 
ohms  so  that  the  capacity  J  of 
the  condenser  required  for 
compensation  is  .0000028 
farad  or  2.8  microfarads. 
The  synchronous  motor  run- 
ning at  light  load  has  negative 
reactance  and  may  be  used  in 
place  of  a  condenser. 
61.  Problem  X.  The  transformer  without  iron. — Given  two  coils 


FURTHER   FUNDAMENTAL  PROBLEMS.  TJ 

of  wire,  Fig.  70,  a  primary  coil  A  and  a  secondary  coil  B  placed 
near  together  but  not  electrically  connected.  The  secondary 
coil  B  is  connected  to  a  closed  circuit  of  which  the  total  resist- 
ance is  r^  and  the  total  inductance  is  L^.  The  resistance  of  the 
primary  coil  is  r^  its  inductance  is  L^  and  the  mutual  inductance 
of  the  two  coils  is  M.  It  is  required  to  find  the  e.  m.  f  which 
must  act  upon  the  terminals  of  the  primary  coil  to  maintain  in 
this  coil  a  given  harmonic  current. 

Determination  of  the  secondary  current  I^. — The  given  primary 
current  induces  in  the  secondary  coil  an  e.  m.  i.  which  is  at  each 

di    * 
instant  equal  to  M  -j  - 

This  e.  m.  f  is  90°  ahead  of  /^  its  effective  value  is  (oMI^,  and 
its  symbolic  expression  isjcoMI^,  so  that  according  to  problem 
IV.  this  e.  m.  f  produces  in  the  secondary  coil  a  current 

1.=^^-  (a) 

Reaction  of  secondary  current  upon  the  primary  coil. — The  sec- 
ondary current  I^  induces  in  the  primary  coil  an  e.  m.  f  which  is 

dt 

in  phase,  its  effective  value  is  oiMI^,  and  its  complex  expression 
is  jcoMI^.  This  e.  m.  f  induced  in  the  primary  by  the  secondary 
current  must  be  overcome  by  the  e.  m.  f  which  acts  upon  the 
primary.  The  portion  of  the  acting  e.  m.  f  which  thus  balances 
the  reaction  of  the  secondary  current  is  equal  to  this  reaction 
and  opposite  to  it  in  sign  and  is,  therefore,  equal  to  —jcoMI^. 

Determination  of  total  e.  in.  f  acting  on  primary. — This  total 
e.  m.  f  consists  of  three  parts  as  follows  : 

(i)  The  part  described  above  which  balances  the  reaction  of 
the  secondary  current.     This  part  is  equal   to  —jcoMI^  or  using 

di 
*The  expression  for  this  induced  e.  m.  f.  is  usually  written  — M  —.;   this  negative 

at 
sign  is,  however,  a  conventional  matter. 


at  each  instant  equal  to  M  -f'-      This  e.  m.  f  is  90°  ahead  of /2 


78 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


the  value  of  /g  from  equation   (a)   we   have  for  this  part  of  the 
total  e.  m.  f. 


^2  +y^A 

(2)  The  part  used  to  overcome  the  resistance  of  the  primary 
coil.  This  is  at  each  instant  equal  to  r^z^,  its  effective  value  is 
rjy,  and  its  complex  expression  is  i\I^  since  it  is  in  phase  with  /^. 

(3)  The  part  used  to  overcome  the  inductance  of  the  primary 

di     .     . 
coil.     This  is  at  each  instant  equal  to  L,  -^>  it  is   qo°  ahead  of 

^  dt 

I^,  its   effective  value  is  (oL^I^,  and  its   complex  expression  is 

jo)LJ^.      Therefore  the  total   e,  m,  f.  required  to  maintain  the 

given  primary  current  is 

.  .2  TX/ri  T 

(b) 


^2  +y"'A 


or  separating  components 


^'  =  [('•'  +  vf^^Zi)  +  J  K-  vfti?L^ )V^     (=) 

Fig.  71  shows  the  primary  current  /, 
the  e.  m.  f.  jioMI^  induced  in  the  sec- 
ondary coil,  the  secondary  current  I^,  the 
portion  of  the  primary  e.  m.  f.  —j'coMI^ 
used  to  balance  the  reaction  of  the  sec- 
ondary current,  the  portion  of  the  pri- 
mary e.  m.  f.  r^/j  used  to  overcome  pri- 
mary resistance,  and  the  portion  of  the 
primary  e.  m.  {.  jcoLJ^  used  to  overcome 
primary  inductance.  The  total  primary 
e.  m.  f.  is  the  vector  sum  oi—jcoMI.^, 
r^I^  diWd  JO) L-^I-^. 

Equation  (c)  shows  that  the  effect  of 

^JcOJiHIii  the  secondary  coil  is  to  make  the  pri- 

FiG.  71.  mary  coil  behave  as  if  its  resistance  were 


increased   by  the    amount 


r'  +  ^o^L,^' 


and    its  inductance  de- 


creased  by  the  amount 


"FURTHER   FUNDAMENTAL   PROBLEMS.  79 


2 


'2     ■^^'^ 

Problems. 

1.  A  transmission  line  having  an  inductance  of  .02  henry 
and  a  resistance  of  2  5  ohms  supplies  current  at  a  frequency  of  60 
cycles  per  second  to  condenser  of  which  the  capacity  is  0.24 
microfarad.  The  e.  m.  f  of  the  alternator  generator  is  20,000 
volts.  Calculate  the  effective  and  maximum  e.  m.  f  at  the  con- 
denser terminals. 

2.  A  transformer  (without  iron)  consists  of  two  long  cylin- 
drical coils  each  having  i  o  turns  of  wire  per  centimeter  of  length 
(one  layer).  The  coils  are  each  50  cm.  long  and  their  radii  are  2 
cm.  and  3  cm.  respectively,  the  smaller  coil  being  inside  the  larger. 
Calculate  the  value  and  phase  of  the  e.  m.  f  required  to  maintain 
a  current  of  10  amperes  at  60  cycles  per  second  in  the  outer  coil  ; 
calculate  the  current  in  the  inner  coil,  and  calculate  the  apparent 
reactance  and  resistance  of  the  outer  coil.  The  outer  coil  has  2 
ohms  resistance  and  the  inner  coil  has  i  y^,  ohms  resistance  and 
its  terminals  are  short  circuited.* 

3.  A  i6-candle  50-volt  lamp  has  about  50  ohms  resistance. 
Such  a  lamp  is  connected  in  series  with  an  inductance  of  2  hen- 
rys,  and  another  is  connected  in  series  with  a  condenser  of  which 
the  capacity  is  3  microfarads.  These  two  circuits  are  connected 
in  parallel,  and  through  a  third  lamp  to  500-volt  125-cycle 
mains.     Calculate  the  current  in  each  lamp. 

*The  mutual  inductance,  in  henrys,  of  two  coaxial  solenoids  is,  approximately, 

M^/^-h'f'^z'z'n—  lo9, 

in  which  z'  and  z'^  are  the  turns  of  wire  per  unit  length  on  the  respective  coils,  r"  is 
the  radius  of  the  inside  coil,  and  /  is  the  length  of  the  coils. 


CHAPTER   VIII. 

SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS.* 

62.  Advantag^es  of  alternating  currents. — When  electric  power 
is  transmitted  over  considerable  distances  the  transmission  wires 
must  be  very  large  to  prevent  excessive  loss,  unless  the  power  is 
transmitted  by  a  small  current  pushed  by  a  high  e.  m.  f.  or  pres- 
sure. Considerations  of  safety  to  life  and  property,  however, 
make  it  inadvisable  to  furnish  power  to  the  user  in  the  form  of 
small  current  at  high  pressure,  so  that  it  is  now  usual  to  trans- 
form the  transmitted  power  and  furnish  it  to  the  user  in  the  form 
of  large  current  at  low  pressure.  The  advantage  of  alternating 
currents  over  direct  currents  lies  mainly  in  the  simplicity  and  com- 
parative cheapness  of  the  alternating  current  apparatusf  required 
to  transform  from  a  high  pressure  and  small  current  to  a  low 
pressure  and  large  current  or  vice  versa.  Alternating  current 
generators  also  are  better  adapted  for  the  generation  of  high  po- 
tential than  direct  current  generators  on  account  of  the  absence 
of  the  commutator  and  the  comparative  simplicity  of  the  arma- 
ture winding  which  admits  of  high  insulation. 

63.  The  single-phase  alternator  and  its  limitations. — The  simple 
alternator  described  in  Chapter  II.  is  called  a  single-phase  alter- 
nator. It  has  one  pair  of  collector  rings  to  which  the  termi- 
nals of  the  armature  winding  are  connected.  The  current  given 
by  a  single-phase  alternator  is  entirely  satisfactory  for  electric 
lighting!  and  in  general  for  all  purposes  in  which  the  heating 

*The  single-phase  alternator  has  been  described  in  Chapter  II.  The  present 
chapter  deals  with  the  essential  features  of  alternators  for  producing  polyphase  cur- 
rents and  Chapter  IX.  treats  in  detail  of  the  theory  and  design  of  single-phase  and 
polyphase  alternators.  The  meaning  of  the  term  polyphase  will  appear  in  the  course 
of  the  following  discussion. 

t  See  Chapter  X.     The  alternating- current  transformer. 

X  Arc  lamps  do  not  operate  quite  as  well  with  alternating  current  as  with  direct 
current. 

(80) 


SINGLE-PHASE   AND    POLYPHASE  ALTERNATORS. 


effect  only  of  the  current  is  important.  For  motive  purposes  the 
simple  alternating  current  is  not  satisfactory,  as  it  is  difficult  to 
make  a  single-phase  alternating-current  motor  which  will  start 
satisfactorily  under  load.  For  electrochemical  processes  the 
alternating  current  cannot  be  used. 

The  satisfactory  use  of  alternating  currents  for  motive  purposes 
depends  mainly  upon  the  use  of  the  induction  motor  described  in 
Chapter  XIII.  It  is  the  requirements  of  this  motor  which  has 
led  to  the  development  of  polyphase  systems. 

64.  The  two-phase  alternator. — Consider  two  similar  and  in- 
dependent single-phase  armatures  A  and  B,  Fig.   72,  mounted 

rigidly  on  the  same  shaft,  one 
beside  the  other,  and  re- 
volved inside  the  same  crown 
of  field  magnet  poles.  In 
the  figure,  armature  B  is 
shown  inside  of  A  for  the 
sake  of  clearness.  These 
armatures  are  so  mounted  on 
the  shaft  that  the  slots  of  A 
are  midway  under  the  poles 
when  the  slots  of  B  are 
midway  between  the  poles  as 
shown.  Under  these  con- 
ditions the  e.  m.  f.'s  of  A  and 
B  are  so  related  in  their  pulsations  that  the  e.  m.  f.  of  A  is  at 
its  maximum  when  the  e.  m.  f.  of  B  is  zero,  that  is  the 
e.  m.  f.'s  are  90°  apart  in  phase,  or  in  quadrature  with  each 
other.  Two  alternators  connected  (mechanically)  in  the  manner 
indicated  constitute  a  two-phase  alternator.  The  two  distinct  and 
independent  e.  m.  f's  generated  by  such  a  machine  are  used  to 
supply  two  distinct  and  independent  currents  to  two  distinct  and 
independent  circuits.  In  practice  the  two-phase  alternator  is 
made  by  placing  the  armature  windings  of  A  and  B  upon  one 

6 


Fig.  72. 


82         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

and  the  same  armature  body.     For  this  purpose  the  armature 
body  has  twice  as  many  slots  as  A  or  B,  Fig.  72.     Fig.  73  shows 


Fig.  73. 

such  an  armature.  The  slots  marked  a^,  a^,  a^,  etc.,  receive  the  con- 
ductors belonging  to  phase  A,  and  those  marked  b^,  b^,  b^,  etc., 
receive  those  belonging  to  phase  B.  The  A  winding  would  pass 
up  *  slot  «j,  down  ^2)  up  <^3  and  so  on,  the  terminals  of  the  wind- 
ing being  connected  to  two  collector  rings.  The  B  winding 
would  pass  up  slot  b^,  down  b^,  up  b^,  and  so  on,  its  terminals 
being  connected  to  two  collector  rings  distinct  from  those  to 
which  the  A  winding  is  connected. 

The  armature  windings  A  and  B  here  described  are  of  the  con- 
centrated type  (see  Art.  19)  having  only  one  slot  per  pole  for 
each  winding.  Distributed  windings  also  are  frequently  used  for 
two-phase  alternators.  Thus  Fig.  74  shows  a  portion  of  a  two- 
phase  armature  with  its  A  and  B  windings  each  distributed  in  two 

*  Up  and  down  being  parallel  to  the  armature  shaft  io  BXid-frotn  one  end  of  the  ar- 
mature. 


SINGLE-PHASE   AND   POLYPHASE  ALTERNATORS. 


83 


The  coils  belonging  to  windings  A  and  B  are 


slots  per  pole 

differently  shaded  to  distinguish  them 


The  coils  belonging  to 


Fig.  74. 

winding  A  are  connected  together  in  a  manner  indicated  by  the 
dotted  lines  and  the  coils  belonging  to  winding  B  are  connected 
together  as  indicated  by  the  full  lines.      [See  Art.  84.] 

In  general  two-phase  alternators  are  provided  with  two  pairs  of 
collector  rings  ;  occasionally  however  one  ring  is  made  to  serve 


urindim\ 


B 

winding 


jnain  I 


jnaiji  s. 


Tzidin,  3 


Fig.  75. 

as  a  common  connection  for  the  two  phases  as  shown  in  Fig.  75. 
65.  Two-phase  e.  m.  f.'s  and  currfents. — The  two  lines  A  and  B, 
Fig.  jd,  represent  the  e.  m.  f.'s  of  the  A  and  B  windings  respect- 
ively of  a  two-phase  alternator.  If  the  circuit  which  receives  cur- 
rent from  A  is  of  the  same  resistance  and  reactance  as  the  circuit 


84         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

which  receives  current  from  B  then  the  system  is  said  to  be  balanced 
and  each  current  lags  behind  its  e.  m.  f.  by  the  same  amount.     In 

this  case  the  currents  are  equal  and  in 
quadrature  with  each  other  and  are  rep- 
resented by  the  two  dotted  lines  a  and  b 
ji  m.  Fig.  76. 

I  •  66.    Electromotive   force    and    current 

-^V  relations  in  two-phase  system.     Electvo- 

I  motive  force. — ^The  e.  m.  f.  between  the 

I  mains  i  and  3,  Fig.  75,  is  the  sum*  of 

J      *  the  e.  m.  f's  A  and  B,   Fig.  ^6.     This 

e.  m.  i.  is  therefore  represented  by  the 


K 


«5^^-, 


Fig.  76.  diagonal  of  the  parallelogram  constructed 

on  A  and  B,  Fig.  yd.  It  is  45  °  behind  A 
in  phase  and  its  effective  value  is  s/  2E  where  E  is  the  common 
effective  value  of  the  e.  m.  f's  A  and  B. 

Current. — The  current  in  main  2  is  the  sum  *  of  the  currents 
in  mains  i  and  3,  namely  a  and  b,  Fig.  'j6.  This  current  is  there- 
fore represented  by  the  diagonal  of  the  parallelogram  constructed 
on  a  and  b,  Fig.  76 ;  it  is  45°  behind  a  in  phase  and  its  effective 
value  is  -v/2/ where  /  is  the  common  effective  value  of  the  cur- 
rents a  and  b. 

67.  The  three-phase  alternator. — Consider  three  similar  single- 
phase  armatures,  A,  B  and  C,  mounted  side  by  side  on  the  same 
shaft  and  revolved  in  the  same  field.  Fix  the  attention  upon  a 
certain  armature  slot  of  A  and  let  time  be  reckoned  from  the  in- 
stant that  this  slot  is  squarely  under  an  iV-pole.  Let  t  be  the  time 
which  elapses  as  this  armature  slot  passes  from  the  center  of  one 
A''-pole  to  the  center  of  the  next  iV-pole.  The  armature  B  is  to 
be  so  fixed  to  the  shaft  that  its  slots  are  squarely  under  the  poles 
at  the  instant  |-  t,  and  the  armature  C  is  to  be  so  fixed  that  its 
slots  are  squarely  under  the  poles  at  the  instant  |-  t.  While  a 
slot  passes  from  the  center  of  one  iV-pole  to  the  center  of  the 

*  Or  difference  according  to  convention  as  to  sign. 


SINGLE-PHASE   AND   POLYPHASE   ALTERNATORS. 


85 


next  iV^-pole  the  e.  m.  f.  passes  through  one  complete  cycle. 
Hence  the  e.  m.  f.'s  given  by  three  armatures,  arranged  as  above, 
will  be  1 20°  apart  in  phase,  as  shown  in  Fig.  'jj,  in  which  the 
lines  A,  B  and  C  repre- 
sent the  respective  e.  m. 
f.'s.  The  currents  given 
by  the  armatures  to  three 
similar  receiving  circuits 
lag  equally  behind  the  re- 
spective e.  m.  f.'s  and  are 
represented  by  the  dot- 
ted lines  a,  b  and  c.  This 
combination  of  three  al- 
ternators is  called  a  three- 
phase  alternator.  In  prac- 
tice the  three  distinct 
windings  A,  B  and  C  are 
placed  upon  one  and  the  same  armature  body.  For  this  purpose 
the  armature  body  has  three  times  as  many  slots  as  A,  B  or  C. 


Fig.  77. 


M-UM 


'  / 


Fig.  78. 


S6 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


Fig.  yS  shows  the  arrangement  of  the  slots  for  such  a  winding. 
The  slots  belonging  to  phase  A  are  drawn  in  heavy  lines  and  are 
marked  a^^,  a^,  etc.  Those  belonging  to  phase  B  are  shown 
dotted  and  those  belonging  to  phase  C  are  shown  in  light  lines. 
The  A  winding  would  pass  up  slot  a^,  down  a^,  up  a^,  etc.;  the 
B  winding,  up  d^,  down  d^,  up  d^,  etc.;  and  similarly  for  phase  C. 


The  windings  A,  B  and  C  here  described  are  of  the  concen- 
trated type,  having  only  one  slot  per  pole  for  each  winding.  Dis- 
tributed windings  also  are  frequently  used  for  three-phase  alterna- 
tors.    Thus  Fig.  79  shows  a  portion  of  a  three-phase  armature 


Fig.  8o. 


with  its  A,  B  and  C  windings  each  distributed  in  two  slots  per 
pole.  The  coils  belonging  to  windings  A,  B  and  C  respectively 
are  differently  shaded  to  distinguish  them.  The  manner  of  con- 
necting the  coils  of  each  winding  is  described  in  Art.  84. 


SINGLE-PHASE   AND    POLYPHASE   ALTERNATORS. 


^7 


Fig.  8i. 


If  the  three  circuits  of  a  three-phase  alternator  are  to  be  en- 
tirely independent  six  collector  rings  must  be  used,  two  for  each 
winding  ;  however,  the  cir- 
cuits may  be  kept  practi- 
cally independent  by  using 
four  collector  rings  and 
four  mains,  as  shown  in 
Fig.  80.  The  main  4 
serves  as  a  common  re- 
turn wire  for  the  indepen- 
dent currents  in  mains  i, 
2  and  3.  When  the  three 
receiving  circuits  are  equal 
in  resistance  and  react- 
ance, that  is,  when  the 
system  is  balanced,  the 
three    currents   are   equal 

and  1 20°  apart  in  phase  (each  current  lagging  behind  its  e.  m.  f.  by 
the  same  amount)  and  their  sum  is  at  each  instant  equal  to  zero  :  in 

which  case  main  4,  Fig.  80, 
carries  no  current  and  this 
main  and  the  corresponding 
collector  ring  may  be  dis- 
pensed with,  the  three  wind- 
ings being  connected  to- 
gether at  the  point  JV,  called 
the  common  junction.  This 
arrangemnnc,  shown  in  the 
symmetrical  diagram,  Fig. 
81,  is  called  the  Y,  or  star, 
scheme  of  connecting  the 
three  windings  A,  B  and  C. 
Another  scheme  for  connecting  the  three  windings  A,  B  and 
C  (also  for  balanced  loads)  called  the  J  (delta)  or  mesh  scheme 


Fig.  82, 


88 


THE  ELEMENTS   OF   ALTERNATING   CURRENTS. 


is  shown  in  Fig.  82.  Winding  A  is  connected  between  rings  3 
and  I,  winding  B  between  rings  i  and  2  and  winding  C  between 
rings  2  and  3. 

The  direction  in  a  circuit  in  which  the  electromotive  force  or 
current  is  considered  as  a  positive  e.  m.  f.  or  current  is  called  the 
positive  direction  through  the  circuit.  This  direction  is  chosen 
arbitrarily.  The  arrows  in  Figs.  81  and  82  indicate  the  positive 
directions  in  the  mains  and  through  the  windings.  It  must  be  re- 
membered that  these  arrows  do  not  represent  the  actual  directions 


Fig.  83. 

of  the  e.  m.  f.'s  or  currents  at  any  given  instant  but  merely  the 
directions  oi positive  e.  m.  f.'s  or  currents.  Thus  in  Fig.  81  the 
currents  are  considered  positive  when  flowing  from  the  common 
junction  towards  the  collecting  rings  and  the  currents  are  never 
all  of  the  same  sign. 

68.  Electromotive  force  and  current  relations  in  Y-connected 
armatures.     E.  in.  f.  relations. — Passing  through  the  windings  A 


SINGLE-PHASE   AND    POLYPHASE  ALTERATORS. 


89 


and  B  from  ring  2  to  ring  i,*  in  Fig.  81  the  winding^  is  passed 
through  in  the  positive  direction  and  the  winding  B  in  the  nega- 
tive direction.  Therefore,  the  e.  m.  f  between  mains  i  and  2  is 
A  —  B  (see  Fig.  yy).  Similarly  the  e.  m.  f  between  mains  2  and 
2,\sB—C  and  the  e.  m.  f.  between  mains  3  and  i  is  C  —  A. 
These  differences  are  shown  in  Fig.  2>T).  The  e.  m.  f.  between 
mains  i  and  2,  namely,  A  —  B,  is  30°  behind  A  in  phase  and  its 
effective  value  is  2^5' cos  30°  =  v'3^,  where  E  is  the  common 
value  of  each  of  the  e.  m.  f.'s,  A,  B  and  C.  Similar  statements 
hold  concerning  the  e.  m.  f.'s  between  mains  2  and  3  and  be- 
tween mains  3  and  i.  Hence  the  e.  m.  f.  between  any  pair  of 
mains  leading  from  a  three-phase  alternator  with  a  K-connected 
armature  is  equal  to  the 

e.  m.  f.  generated  per  phase  ^c, 

multiplied  by  \/3.  ^^'^ 

Current  relations.  —  In 
the  F  connection  the  cur- 
rents in  the  mains  are 
equal  to  the  currents  in 
the  respective  windings,  ^^— 
as  is  evident  from  Fig.  8 1 . 

69.  Electromotive  force 
and  current  relations  in  J 
connected  armatures.  E. 
m.  f.  relations. — In  //  con- 
nected armatures  the  e.  m. 
f.'s  between  the  mains  or 
collector  rings  are  equal 
to  the  e.  m.  f 's  of  the  respective  windings  as  is  evident  from 
Fig.  82. 

Current  relations. — Referring  to  Fig.  82  we  see  that  a  positive 
current  in  winding  A  produces  a  positive  current  in  main  i  and 

*  Which  is  the  direction  in  which  an  e.  m.  f.  must  be  generated  to  give  an  e.  m.  f.> 
acting  upon  a  receiving  circuit  from  main  I  to  main  2. 


Fig.  84. 


go         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

that  a  negative  current  in  winding  B  produces  a  positive  current 
in  main  i,  therefore  the  current  in  main  i  is  a-b  when  a  is  the 
current  in  A  and  b  is  the  current  in  B.  Similarly  the  current  in 
main  2  is  b-c  and  the  current  in  main  3  is  c-a.  These  differences 
are  shown  in  Fig.  84.  The  current  in  i,  namely  a-b,  is  30° 
behind  a  in  phase  and  its  effective  value  \s  V  t^  I  when  /  is  the 
common  effective  value  of  the  currents  a,  b,  c  in  the  different 
phases.  Similar  statements  hold  for  the  currents  in  mains  2  and 
3  ;  so  that  the  current  in  each  main  of  a  J  connected  armature 
is  y  -iy  times  the  current  in  each  winding. 

70.  Connection  of  receiving  circuits  to  three-phase  mains.  In 
case  of  dissimilar  circuits  (unbalanced  system). — When  the  re- 
ceiving circuits  which  take  current  from  three-phase  mains  are 
dissimilar,  four  mains  should  be  employed  as  indicated  in  Fig. 
80 ;  each  receiving  circuit  being  connected  from  main  4  to  one 
of  the  other  mains.  It  is  however  desirable  to  keep  the  three 
windings  A,  B  and  C  of  the  alternator  as  nearly  equally  loaded 
as  possible,  and  the  receiving  circuits  are  so  disposed  in  practice 
as  to  satisfy  this  condition  as  nearly  as  possible. 

/;/  case  of  similar  circuits  (balanced  system). — When  three- 
phase  currents  are  used  to  drive  induction  motors,  synchro- 
nous   motors    or    rotary    converters,    each    unit    takes    current 

equally    from    the    three 

J      .         ^,  main  1 

mams,    and    smce    three- 

phase  currents  are  utiHzed 
mainly  in  the  operation  of 
the  machines  mentioned, 
the  system  is  usually  bal- 
anced. In  this  case  three 
mains  only  are  employed 
and  each  receiving  unit 
has  three  similar  receiving  ^^'    ^' 

circuits  connected  to  the  mains  according  to  either  the  For  z/ 
method.     The    Y  method   of   connecting   receiving   circuits    is 


SINGLE-PHASE   AND   POLYPHASE   ALTERNATORS.        9 1 

shown  in  Fig.  85.  One  terminal  of  each  receiving  circuit  is  con- 
nected to  a  main  and  the  other  terminals  are  connected  together 
at  N.  In  this  case  the  current  in  each  receiving  circuit  is  equal 
to  the  current  in  the  main  to  which  it  is  connected.  The  e.  m.  f. 
between  the  terminals  of  each 
receiving   circuit  is  equal  to  main  / 

E 
— =  where  E  is  the  e.  m.  f  . 

...  main  z      J^  ^a 

between  any  pair  of  mams.  ~  ^ 

The  A  method  of  connect- 
ing receiving  circuits  is  shown  main  5 


in  Fig.  ^6.   Here  the  three  re-  Fig.  86. 

ceivinpf  circuits  are  connected 

between  the  respective  pairs  of  mains,  the  e.  m.  f.  acting  on  each 

receiving  circuit  is  the  e.  m.  f.  between  the  mains,  and  the  current  in 

each  receiving  circuit  is — ^  where  /is  the  current  in  each  main. 

71.  Power  in  polyphase  systems. — The  several  circuits  of  a 
polyphase  system  are  in  general  entirely  separate  and  independ- 
ent, and  the  total  power  delivered  to  a  receiving  apparatus  is  to 
be  found  by  measuring  the  power  delivered  to  each  separate  re- 
ceiving circuit ;  the  total  power  delivered  is  the  sum  of  the 
amounts  delivered  to  the  different  receiving  circuits. 

Balanced  systems. — When  a  polyphase  system  is  balanced  the 
several  circuits  are  entirely  similar  and  the  same  amount  of  power 
is  delivered  to  each  receiving  circuit  of  a  given  piece  of  receiving 
apparatus. 

Balanced  two  phase. — Let  E  be  the  e.  m.  f.  of  each  phase,  / 

the  current  furnished  to  each  of  two  similar  receiving  circuits, 

and   cos  d   the   power   factor  of  each    receiving   circuit.     Then 

EI  cos  d  is  the  power  delivered  to  each  circuit  so  that  the  total 

power  delivered  is 

P=2EIcosd.  (55) 

Balanced  three  phase. — Let  E  be  the  e.  m.  f.  between  the  ter- 


92         THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 

minals  of  each  receiving  circuit,  /  the  current  in  each  receiving 
circuit,  and  cos  d  the  power  factor  of  each  circuit.  Then  EI 
cos  d  is  the  power  dehvered  to  each  receiving  circuit  so  that  the 
total  power  delivered  is 

P=  ^  EI  cos  d  (56) 

in  which,  as  must  be  remembered,  E  is  the  e.  m.  f.  at  the  ter- 
minals of  each  receiving  circuit  and  /  is  the  current  in  each  re- 
ceiving circuit.     On  the  other  hand  : 

P  =  V^  EI  cos  d  (57) 

in  which  E  is  the  e.  m.  f.  between  each  pair  of  mains,  /  is  the 
current  in  each  main,  and  cos  6  is  the  power  factor  of  each  re- 
ceiving circuit.  Equation  (57)  may  be  derived  from  (56)  by- 
considering-  that  the  current  /  in  each  main,  for  the  case  of  // 
connection  for  example,  is   equal  to  n/3  /,  so  that,  substituting 

—^  for  /in  equation  (56)  we  have  equation  (57). 

72.  The  flow  of  energy  in  balanced  polyphase  systems. — It  was 

pointed  out  in  Art.  22  that  the  power  developed  by  a  single-phase 
alternator  pulsates  with  the  alternations  of  e.  m.  f.  and  current. 
The  power  delivered  to  a  balanced  system  by  a  polyphase  gener- 
ator, on  the  other  hand,  is  not  subject  to  pulsations  but  is  entirely 
steady  and  constant  in  value. 

Discussion  for  a  two-phase  alternator :      Consider  a  single-phase  alternatol j  of 
which  the  e.  m.  f.  is  : 

^  =  J^sin6)/  (a) 

and  which  gives  a  current 

i^Isax  [ut — 6) 

or  2  =  J  sin  ut  cos  6  —  J  cos  ut  sin  6.  (b) 

The  instantaneous  power  ei  is  : 

ei  =  ^I  cos  6  sin2  ut  —  J^I  sin  6  sin  ut  cos  ut 

which  pulsates  with  a  frequency  twice  as  great  as  the  frequency  of  e  and  i. 

Let  equations  (a)  and  (b)  express  the  e.  m.  f.  and  current  of  one  phase  of  a  (bal- 
anced) two-phase  alternator,  then  the  e.  m.  f.  and  current  of  the  other  phase  are 

^''  =  J^cosw/  (c) 

i''  =  J  cos  {ut  —  6)  =1  cos  ut  cos  6  -\- 1  sin  ut  sin  6.  ( d ) 


SINGLE-PHASE  AND   POLYPHASE   ALTERNATORS. 


93 


The  instantaneous  power  output  of  this  phase  is 

e'i'  =  J5  J  cos  Q  cos2  ut  -\-  JB^I  sin  Q  sin  w/  cos  w^. 

Therefore  the  total  power  output  of  the  two-phase  machine  is 

ei  -f-  e'i'  =-  ^I  cos  6  (sin^  u(  -\-  cos^  ut) 

^:E^I  cos  d 
which  is  constant. 

Remaj^k:  The  torque  of  a  single-phase  alternator  pulsates  with 
the  pulsations  of  the  power  output.  In  a  balanced  polyphase  al- 
ternator however  the  torque  is  steady  since  the  power  does  not 
pulsate  ;  also  polyphase  synchronous  motors,  rotary  converters, 
and  induction  motors  are  driven  by  a  steady  torque.  ■ 

73.  Measurement  of  power  in  polyphase  systems. — In  a  poly- 
phase system,  balanced  or  unbalanced,  the  power  taken  by  any 
unit,  such  as  an  induction  motor,  maybe  determined  by  measure- 
ing  the  power  taken  by  each  single  receiving  circuit  and  adding 
the  results.  In  order  to  measure  the  power  taken  by  a  single 
receiving  circuit  the  current  coil  of  the  wattmeter  is  connected  in 
series  with  the  circuit  and  the  fine  wire  coil  is  connected  to  the 
terminals  of  the  circuit.     The  inconvenience  of  connecting  and . 


Fig.  87. 

disconnecting  the  wattmeters  makes  it  necessary  to  use  a  separate 
wattmeter  for  each  receiving  circuit.  Two  wattmeters  are  suffi- 
cient for  the  complete  measurements  of  the  power  taken  by  any 
three-phase  receiving  unit.     The  connections  are  shown  in  Fig. 


94         THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

^y.     The  receiving  circuits  may  be  balanced  or  unbalanced  and 
connected  Vox  J. 

In  a  balanced  polyphase  system,  a  condition  which  is  seldom 
strictly  realized,  the  power  taken  by  one  only,  of  the  receiving 
circuits  need  be  measured. 

Problems. 

1.  A  common  return  wire  is  used  for  the  two  currents  of  a  two- 
phase  system.  The  system  is  balanced  and  each  current  is  equal 
to  lOO  amperes.  What  is  the  current  in  the  common  return 
wire  ? 

2.  The  e.  m.  f  of  each  phase,  problem  i,  is  500  volts.  What 
is  the  e.  m.  f  between  the  outside  wires  ? 

3.  Three  similar  receiving  circuits  are  A  connected  to  3 -phase 
mains,  the  e.  m.  f  between  each  pair  of  mains  being  no  volts. 
The  power  delivered  to  the  three  circuits  is  1 50  kilowatts  and  the 
power  factor  of  each  circuit  is  .90.  What  is  the  current  in  each 
circuit  and  in  each  main  ? 

4.  Three  similar  receiving  circuits  are  Y  connected  to  the  3- 
phase  mains,  problem  3  ;  the  total  power  delivered  is  150  kilo- 
watts and  the  power  factor  of  each  circuit  is  .90.  What  is  the 
current  in  each  circuit  and  in  each  main ;  and  what  is  the  e.  m.  f. 
between  the  terminals  of  each  circuit  ? 


CHAPTER   IX. 

ALTERNATORS. 
( Continued. ) 

74.  Armature  reaction. — The  amount  of  magnetic  flux  enter- 
ing the  armature  core  from  the  field  poles  and  the  manner  of  its 
distribution  over  the  pole  faces,  depend  upon  the  combined  mag- 
netic action  of  the  field  current  and  of  the  armature  current. 

Distortion  of  field. — The  armature  current  in  an  alternator 
tends  to  concentrate  the  magnetic  flux  under  the  trailing  horns 
of  the  pole  pieces  very  much  as  in  the  direct  current  dynamo. 
The  effect  of  this  concentration  of  flux  is  to  slightly  increase  the 
magnetic  reluctance  of  the  air  gap  and  of  the  saturated  portions 
of  the  pole  pieces  and  armature  core.  This  increase  of  magnetic 
reluctance  causes  a  decrease  of  flux  and  a  consequent  decrease  of 
the  e.  m.  f.  of  the  alternator,  others  things  being  equal.  This 
effect  may  be  appreciable,  but  it  cannot  be  accurately  calculated. 

Magnetizing  and  demagnetizing  action  of  the  armature  current. 
— When  the  current  given  by  an  alternator  is  in  phase  with  its 
e.  m.  f.  the  only  effect  of  the  armature  current  upon  the  field  is 
the  distorting  effect  described  above.  When  the  current  is  not 
in  phase  with  the  e.  m.  f  the  distorting  effect  is  decreased*  and 
in  addition  there  is  a  magnetizing  action  or  demagnetizing  action 
upon  the  field  according  as  the  current  is  ahead  of  or  behind  the 

e.  m.  f  in  phase. 

Consider  a  bundle  of  Z  armature  wires,  grouped  in  a  slot,  for  example.      Let 

e  =  ^  sin  (j)t. 
be  the  alternating  e.  m.  f.   induced  in  this   bundle  of  conductors.      This  e.  m.  f. 
is  a  maximum  when  the  slot  is  at  a,  Fig.   88.     It  is  zero  when    the  slot  is  at  t> 

*  The  distorting  effect  is  due  to  the  component  of  the  current  parallel  to  the  e.  m. 

f.  and  the  magnetizing  effect  is  due  to  the  component  at  right  angles  to  the  e,  m.  f. 

(95) 


96 


THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


and  it  is  a  minimum  (negative  maximum)  at  c.     Therefore  the  value  of  ut  is  90** 
at  a,  180°  at  b  and  270°  at  c.      Let  i  be    a  given  current  flowing  in  the  bundle  of 

wires.  The  ampere-turns  of  the  bundle  is 
then  Zi.  If  the  bundle  of  wires  is  at  a  its 
ampere-turns  will  be  .without  appreciable 
effect  on  the  magnetic  circuit  m  m  711  shown 
by  the  dotted  line ;  at  b  the  ampere-turns 
will  have  their  full  demagnetizing  effect 
(negative)*  upon  the  magnetic  circuit,  and 
at  c  the  effect  will  again  be  zei'o. 

Now,  cos  ut  is  zero  at  a,  negative  unity  at  b 
and  zero  at  c.  Therefore  Zi  cos  ut  is  an  ex- 
pression which  gives  the  true  magnetic  effect 
of  the  bundle  of  wires  at  a,  b  and  c.  We 
asszime  his  expression  to  hold  for  all  points 
between  a  and  c.  The  actual  current  in  the 
bundle  of  wires  is 

2  =  J  sin  (w^ — d) 

in  which  ff  is  the  angle  of  lag  of  the  current 
behind  the  e.  m.  f.  Therefore  substituting 
this  value  of  i  in  the  expression  Zi  cos  ut 
we  have 

m  =  ZT  cos  w/  sin  (w/ —  ff). 

in  which  m  is  the  effective  ampere-turns  of  the  bundle  of  wires  at  the  instant  /.     To 
find  the  average  value  of  m  expand  sin  {ji^t  —  0),  whence 

m  =  ZT  sin  ut  cos  tdt  cos  B  —  ZI  cos^  w/  sin  Q. 
Now  the  average  value  of  m  is  the  sum  of  the  average  values  of  the  two  terms  of 
the  right  hand  member  ;  but  the  average  value  of  sin  ut  cos  ut  is  zero  and  the  average 
value  of  cos^  ut\s  y^.     Therefore 

average  ?;?  =  —  JS^  ZI  sin  d 
or  putting  y  2  /for  J  we  have 

average  magnetizing  ampere-turns^ —  .707  Z/sin  d.  (58) 

When  this  formula  is  applied  to  an  alternator  the  windings  of  which  are  not  too 
widely  distributed,  Z  is  to  be  taken  as  the  total  number  of  armature  conductors  divided 
by  the  number  of  poles.  This  equation  shows  that  when  the  current  lags  behind  the 
e.  m.  f.  (angle  Q  positive)  the  armature  current  weakens  the  field  and  vice  versa. 

75.  Armature  inductance. — The  effect  of  the  inductance  of  the 
armature  of  an  alternator  is  to  cause  the  e.  m.  f  between  the  col- 


FlG,  %%. 


*The  current  e  is  considered  positive  when  it  is  in  the  direction  of  the  e.  m.  f. 
which  is  induced  under  the  TV  pole.  A  current  in  this  direction  has  a  demagnetizing 
effect  for  all  positions  of  the  slot  between  a  and  c. 


ALTERNATORS.  9/ 

lecting  rings  to  fall  off  considerably  with  increasing  current, 
especially  if  the  receiving  circuit  is  inductive,  as  is  explained  in 
the  following  article. 

The  inductance  of  an  alternator  armature  is  especially  great  if 
the  conductors  are  deeply  imbedded  in  the  armature  core,  and 
the  inductance  is  increased  by  the  grouping  of  the  conductors 
together  in  bundles.  This  is  evident  from  equation  (7)  which 
expresses  the  inductance  of  a  bundle  of  Z  wires  surrounded  by 
iron.  This  equation  (7)  shows  that  inductance  is  proportional  to 
the  square  of  the  number  of  turns. 

76.  The  electromotive  force  lost  in  the   armature.     Armature 
drop. — The  e,  m.  f  at  the  collecting  rings  of  an  alternator  is  less 
than  the  total  e.  m.  f  in- 
duced in  the  armature  for  y^^ 
the  reason  that  a  portion  of                                      yr  koX  J 
the  induced  e.  m.  f.  is  used 
to  overcome  the  resistance      , 
and  a  portion  is  also  used      ' 
to  overcome  the  inductance 
of  the  armature  windings.                                P^^  g 

General  case. — Let  /, 
Fig.  89,  be  the  current  given  by  the  alternator  and  E  the  total 
induced  e.  m.  f  Then  coLI  is  the  portion  of  E  used  to  over- 
come the  inductance  L  of  the  armature,  and  RI  is  the  portion  of 
E  used  to  overcome  the  resistance  of  R  of  the  armature.  Sub- 
tracting a>Z/and  RI  from  E  gives  the  e.  m.  f  at  the  collecting 
rings,  or  "external  e.  m.  f,"  E^. 

Armature  drop,  non-inductive  load. — In  this  case  E  is  nearly  in 
phase  with  /and  the  subtraction  of  ft>Z/ from  ^  scarcely  reduces 
its  value,  oiLI  being  nearly  at  right  angles  to  E.  Therefore  with 
a  non-inductive  receiving  circuit  the  armature  drop  depends  al- 
most wholly  upon  the  armature  resistance. 

Armature  drop,  inductive  load. — When  the  phase  difference  be- 
tween E and  /is  nearly  90°,  then  the  subtraction  of  RI  from  E 

7 


98 


THE   ELEMENTS    OF   ALTERNATING   CURRENTS. 


scarcely  reduces  its  value.  Therefore  with  a  highly  inductive 
receiving  circuit  the  armature  drop  depends  almost  wholly  upon 
the  armature  inductance. 

77.  The  characteristic  curve  of  the  alternator. — The  curve  ob- 
tained by  plotting  observed  values  of  the  external  e.  m.  f.  for 

various  currents  taken  from 

2400- 
2200. 


■^2000 

^  1800 
•S 1600 

^  1400 
<3 
81200 

I  1000 

•^    800 

Uj   400 
200 


■^ 

' ' 

^ 

"^ 

\ 

^ 

\ 

v^ 

\ 

\ 

\ 

\ 

\ 

5     10 


15    20    25  30    35  40  45   50 

curreni 
Fig.  90. 


an  alternator,  is  called  the 
characteristic  curve  of  the 
alternator.  Such  character- 
istic curves  are  shown  in  Fig. 
90.  Curve  A  is  for  a  sepa- 
rately excited  alternator  hav- 
ing but  small  armature  induc- 
tance and  curve  B  is  for  a 
separately  excited  alternator 
having  large  armature  induc- 
tance. The  shape  of  the 
characteristic  curve  of  a 
given  alternator  depends  to  a 
greater  or  less  extent  upon 
the  inductance  of  the  receiving  circuit.  The  falling  off  of  e.  m.  f 
with  increase  of  current  is  due  in  part  to  the  demagnetizing  action 
of  the  armature  current  which  weakens  the 
field,  and  in  part  to  the  increased  armature 
drop  with  increase  of  current. 

78.  The  constant  current  alternator. — An 
alternator  of  which  the  armature  has  an  ex- 
cessive inductacne,  or  an  ordinary  alternator 
in  circuit  with  which  a  large  inductance  is 
connected,  gives  a  current  which  does  not 
vary  greatly  with  the  resistance*  of  the  re- 
ceiving circuit. 

This  may  be  shown  as  follows  :    Let  E, 


Fjg.  91. 


Fig.  91,  be  the  total  induced  e.  m.  f   of  an  alternator  sending 


*  Unless  the  resistance  becomes  very  large. 


ALTERNATORS. 


99 


current  through  a  circuit  of  which  the  reactance  o)L  is  constant 
and  large,  compared  with  the  resistance  R.  Then  (oLI  will  be 
large,  compared  with  RT.  Further,  RI  and  coLI  are  at  right 
angles  to  each  other  and  their  vector  sum  is  E,  so  that  the  point 
Py  Fig.  91,  lies  on  a  semicircle  constructed  on  j£"  as  a  diameter. 
Now,  when  RI  is  small,  compared  with  E,  then  wLI  is  very 
nearly  equal  to  E,  that  is,  (oLI  is  approximately  constant  and, 
therefore,  /  is  approximately  constant. 


Fig.  92. 

79.  Effect  of  distributed  winding  upon  the  e.  m.  f.  of  an  alter- 
nator.*— Consider  an  armature  winding,  A,  concentrated  in  a  set 

*  This  question  is  discussed  in  a  slightly  different  manner  in  the  chapter  on  the 
rotary  converter. 


lOO       THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 

of  slots,  one  slot  per  pole.    The  effective  e.  m.  f.  of  this  winding  is 

^^  _  4-44  NTf 

according  to  equation  (22),  Chap.  II.  Suppose  another  similar 
concentrated  winding,  B,  is  placed  upon  the  same  armature  in 
slots  distant  s  from  the  first  set  of  slots,  s  being  the  angle  shown 
in  Fig.  92.  This  figure  shows  one  slot  only  of  the  first  set  and 
one  slot  only  of  the  second  set.     The  phase  difference  between 

s 
the  e.  m.  f 's  in  these  two  windings  is  the  angle  -  x  360°,  inas- 
much as  the  angle  q  from  iV  to  iV  is  equivalent  to  360°  of  phase 
difference.     These  two  e.  m.  f 's  are  represented  by  the  lines  A 


___J1 -^ 


and  B^  Fig.  93.  Similarly  the  lines  C  and  D  represent  the  e.  m. 
f.'s  in  two  additional  similar  windings  concentrated  in  two  addi- 
tional sets  of  slots  c  and  d,  Fig.  92.  If  all  these  windings  are 
connected  in  series  the  effective  e.  m.  f  produced  will  be  the 
vector  sum  E  of  A,  B,  C  and  D.  If  we  were  to  calculate  the 
effective  e.  m.  f  produced  by  A,  B,  C  and  D  in  series  on  the  as- 
sumption that  all  the  windings  are  concentrated  in  one  set  of 
slots,  that  is,  if  we  were  to  calculate  this  total  e.  m.  i.  by  means 
of  equation  (22),  using  for  T  the  total  number  of  turns  in  all  the 
windings,  we  would  get  a  result  greater  than  E  in  the  ratio  of  the 
sum  of  the  sides  A,  B,  C  and  D  of  the  polygon  to  the  chord  E, 
Fig-  93-     This  ratio  may  be  called  the  phase  constant  of  the  dis- 


ALTERNATORS. 


lOI 


tributed  winding.     By  introducing  the  phase  constant  k  in  equa- 
tion (22)  this  equation  becomes 

AAAkNTf 

(59) 


E  = 


10° 


This  form  of  the  fundamental  equation  of  the  alternator  is  ap- 
plicable to  armatures  with  distributed  windings.  The  following 
table  gives  the  values  of  k  for  various  degrees  of  distribution. 
The  slots  for  a  given  winding  are  always  grouped  so  many  per 
pole  and  a  group   of  slots  may   cover    y^,  ^,  ^,  etc.,  of  the 

Values  of  Phase  Constant  k  for  Distributed  Windings. 


Widths  of  groups  of  slots  in  fractional  parts  of  N  to  S. 

Number  of  slots 
in  each  Group. 

X 

K 

% 

% 

Whole. 

I 

1.000 

1. 000 

1. 000 

1. 000 

1. 000 

2 

.980 

.966 

.924 

.831 

.707 

3 

.977 

.960 

.912 

.80s 

.666 

4 

.976 

•958 

.908 

•795 

•653 

Infinity. 

.975 

•955 

.901 

.784 

•637 

Note  :  Column  headed  ^  applies  to  3-phase  alternators. 
Column  headed  y^  applies  to  2-phase  alternators. 

Width  of  group  =  wj-,  where  n  is  number  of  slots  in  a  group  and  s  is  distance 
from  center  to  center  of  adjacent  slots. 

space  from  the  center  of  an  iV^pole  to  the  center  of  an  s  pole.  Thus 
in  Fig.  94  is  shown  an  8-pole  machine  of  which  the  armature  is 
slotted  for  a  distributed  winding,  there  being  three  slots  per  pole, 
these  slots  being  grouped  so  as  to  cover  ^  of  the  space  from 
an  iVto  an  vS  pole.  In  the  table  the  width  of  a  group  of  slots  is 
ns  where  n  is  the  number  of  slots  in  a  group  and  s  is  the  dis- 
tance from  center  to  center  of  adjacent  slots. 

80.  Practical  and  ultimate  limits  of  output. — The  dotted  curve, 
Fig-  95,  is  the  characteristic  curve  of  a  given  alternator.  This 
curve  shows  the  relation  between  the  current  output  and  the  e, 
m.  f  between  the  collecting  rings,  the  field  excitation  being  kept 


I02      THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 


constant.     The  ordinates   of  the  full  line  curve   represent  the 
power  outputs  corresponding  to  the  different  currents  (receiving 


26001 


Fig.  94. 

circuit  non-inductive).     The  maximum  output  of  the  alternator 
is  thus  68  kilowatts  when  the  current  output  is  38  amperes.     In 

practice  the  allowable  power 
output  of  an  alternator  is 
limited  to  a  smaller  value 
than  this  maximum  output 
by  one  or  the  other  of  the 
following  considerations. 

{a)  Electric  lighting  and 
power  service  usually  de- 
mands an  approximately 
constant  e.  m.  f.  and  it  is 
not  permissible  to  take  from 
an  alternator  so  large  a  cur- 
rent as  to  greatly  reduce  its 

curreir  "^  ^°  ^'  '"         ^-  ^-  ^-     ^^'^  difficulty  may 
Fig.  95.  be  largely  overcome  by  pro- 


ALTERNATORS.  IO3 

viding  for  an  increase  of  field  excitation  of  the  alternator  with 
increase  of  load  as  is  done  in  the  alternator  with  a  compound 
field  winding.     See  Art.  89. 

{b)  The  current  delivered  by  an  alternator  generates  heat* 
in  the  armature  of  the  alternator  and  the  temperature  of  the  ar- 
mature rises  until  it  radiates  heat  as  fast  as  heat  is  generated 
in  it  by  the  current.  Excessive  heating  of  the  armature  en- 
dangers the  insulation  of  the  windings  and  it  is  not  permissible 
to  take  from  an  alternator  so  large  a  current  as  to  heat  its  arma- 
ture more  than  40°  or  50°  C.  above  the  temperature  of  the  sur- 
rounding air.  This  heating  effect  of  the  armature  currents  usually 
fixes  the  allowable  output  of  an  alternator  except  in  those  rare 
cases  where  extreme  steadiness  of  e.  m.  f  is  required,  or  where 
the  alternator  is  not  compounded. 

Influence  of  inductance  upon  output. — ^An  alternator  is  rated 
according  to  the  power  it  can  deliver  steadily  to  a  non-inductive 
receiving  circuit  without  overheating.  The  amount  of  power 
which  an  alternator  can  satisfactorily  deliver  to  an  inductive  re- 
ceiving circuit  is  less  than  that  which  it  can  deliver  to  a  non-in- 
ductive receiving  circuit,  because  of  the  phase  difference  of  e.  m. 
i.  and  current.  The  cosine  of  this  phase  difference  (cos  6)  is 
called  the  power  factor  of  the  receiving  circuit  as  before  pointed 
out.     The  power  factor  of  lighting  circuits  is  very  nearly  unity. 

The  power  factor  of  induction  motors,  synchronous  motors  and 
rotary  converters  is  often  as  low  as  .75  and  sometimes  even  less. 

81.  Frequencies. — The  frequencies  employed  in  practice  range 
from  20  or  25  to  1 50  cycles  per  second.  Very  low  frequencies  are 
not  suitable  for  lighting  on  account  of  the  tendency  to  produce 
flickering  of  the  lights  ;  on  the  other  hand  high  frequencies,  which 
tend  to  make  transformers  cheaper  for  a  given  output,  are  entirely 
satisfactory  and  are  often  employed  for  lighting. 

High  frequencies  are  not  well  adapted  for  the  operation  of  in- 

*  Additional  heat  is  generated  in  the  armature  by  the  hysteresis  and  eddy  ctirrent 
losses  in  the  armaitore  core. 


I04      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

duction  motors,  synchronous  motors  and  rotary  converters  be- 
cause high  frequencies  necessitate  either  great  speed  or  a  great 
number  of  poles.  For  such  purposes  frequencies  as  low  as  25 
per  second  are  often  employed. 

A  frequency  of  60  has  been  quite  generally  adopted  for 
machines  used  to  operate  both  lights  and  motors. 

82.  Speeds.  Wumljer  of  poles. — A  machine  which  is  to  be  belt- 
driven  may  be  driven  as  fast  as  is  compatible  with  the  strength  and 
rigidity  of  the  rotating  part.  The  allowable  speed  of  rotation  in 
ordinary  dynamos  and  alternators  is  such  as  will  give  a  periph- 
eral velocity  of  from  4000  to  6000  feet  per  minute.  When  a 
machine  is  direct-connected  to  an  engine  or  water  wheel  its  speed 
is  fixed  by  that  of  the  prime  mover. 

The  number  of  poles  depends  upon  the  speed  of  an  alternator 
and  the  frequency  it  is  to  give,  according  to  equation  (18).  Large 
machines  as  a  rule  must  run  slower  than  small  ones  and  they, 
therefore,  have  a  greater  number  of  poles.  The  accompanying 
table  gives  data  as  to  speed,  frequency,  and  number  of  poles  of  a 
few  recent  American  machines.  Machines  7  and  8  are  of  the 
direct-connected  type. 

Table. 


125 

Cycles. 

60  Cycles. 

No. 

No  of 

Output 

Speed 

No. 

No.  of 

Output 

Speed 

poles. 

K.  W. 

r.  p.  m. 

poles. 

K.  W. 

r.  p.  m. 

I 

10 

60 

1500 

4 

8 

75 

900 

2 

14 

125 

1070 

5 

12 

150 

600 

3 

16 

200 

937 

6 

16 

250 

450 

7 

36 

250 

200 

8 

40 

750 

180 

83.  Armatures. — Alternator  armatures  are  usually  of  the  drum 
type  or  disc  type.  The  former  type  is  almost  universal  in  Amer- 
ica while  the  disc  type  is  frequently  used  in  England ;  for  ex- 
ample, the  Ferranti  and  Mordey  machines  have  disc  armatures 


ALTERNATORS.  105 

Drum  armatures  have  laminated  iron  cores  similar  to  the  arma- 
ture cores  used  for  direct-current  dynamos,  while  disc  armatures 
are  usually  made  up  without  iron.  Ring  armatures  have  been 
used  only  to  a  very  limited  extent. 

Drum  armatures  are,  in  nearly  all  modern  machines,  of  the 
toothed  or  ironclad  type.  The  conductors  are  bedded  in  slots. 
This  has  the  double  advantage  of  shortening  the  gap  space  from 
pole  face  to  armature  core  and  of  protecting  the  armature  con- 
ductors from  injury.  One  type  of  such  an  armature  has  already 
been  shown  in  Fig.  10,  the  heavy  coils  being  first  wound  on  forms 
and  then  pressed  into  position  on  the  armature  core.  When  dis- 
tributed windings  are  used  straight  slots  as  shown  in  Fig.  96  are 


Fig.  96.  Fig.  97. 

often  employed.  Fig.  97  shows  a  style  of  slot  commonly  used 
in  which  the  coils  are  held  in  position  by  the  wooden  wedge  JV. 
Armature  core  discs  should  be  varnished,  japanned  or  in  some 
manner  insulated  from  each  other  to  prevent  eddy  currents.  This 
is  especially  necessary  in  the  case  of  alternator  armature  cores 
because  the  frequency  is  comparatively  high. 

84.  Armature  wmdings. — Any  direct  current  dynamo*  may  be 
converted  into  a  single-phase  or  polyphase  alternator  by  provid- 
ing it  with  collecting  rings  as  explained  in  the  chapter  on  the  ro- 
tary converter.  Ordinarily,  however,  the  armature  windings  of 
alternators  are  very  different  from  the  armature  windings  of  di- 
rect-current dynamos.  In  the  type  of  winding  most  frequently 
employed  a  number  of  distinct  coils  are  arranged  on  the  arma- 
ture ;  in  these  coils  alternating  e.  m.  f 's  are  induced  as  they  pass 

*  Except  the  so-called  unipolar  dynamo. 


I06      THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 


the  field  magnet  poles,  and  these  coils  are  connected  in  series  be- 
tween the  collecting  rings  if  high  e.  m.  f.  is  desired  or  in  parallel* 
between  the  collecting  rings  if  low  e.  m.  f.  is  desired. 

Single-phase  winding. — Fig.  i  o  shows  a  common  type  of  single- 
phase  winding  having  one  coil  per  pole.     Fig.  98  shows  another 


Fig.   98. 

type  of  concentrated  single-phase  winding  having  one  coil  to 
each  pair  of  poles  or  one  slot  per  pole.  In  the  diagram,  Fig.  98, 
the  heavy  sector  shaped  figures  represent  the  coils  and  the  light 
lines  represent  the  connections  between  the  terminals  of  the  coils. 
The  radial  parts  of  the  sector  shaped  figures  represent  the  por- 
tions of  the  coils  which  lie  in  the  slots,  and  the  curved  parts  rep- 

*The  coils  of  a  distributed  winding  cannot  all  be  connected  in  parallel  between 
the  collecting  rings  for  the  reason  that  the  induced  e.  m.  f.'s  in  the  various  coils  are 
not  exactly  in  phase  and  local  currents  would  circulate  in  the  coils  if  connected  in 
parallel. 


ALTERNATORS. 


107 


resent  the  ends  of  the  coils.  The  circles  at  the  center  of  the 
figure  represent  the  collecting  rings,  one  being  shown  inside  the 
other  for  clearness.  The  arrows  represent  the  direction  of  the 
current  at  a  given  instant.  All  e.  m.  f  's  under  N  poles  are  in 
one  direction  and  all  e.  m.  f 's  induced  under  vS  poles  are  in  the 
opposite  direction.  These  remarks  apply  to  Figs.  98  to  105  in- 
clusive.    Fig.  99  represents  a  single-phase  winding  distributed  in 


two  slots  per  pole,  all  the  coils  being  connected  in  series.  Fig. 
99  is  a  type  of  winding  which,  for  the  same  number  of  conduc- 
tors, has  a  smaller  inductance  than  the  type  shown  in  Fig.  98  and 
the  armature  shown  in  Fig.  99,  for  the  same  number  of  conduc- 
tors, gives  a  smaller  e.  m.  f  than  the  armature  shown  in  Fig.  98. 
Two-phase  windings. — The  two-phase  winding  is  two  inde- 
pendent single-phase  windings  on  the  same  armature,  each  being 


I08      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

connected  to  a  separate  pair  of  collecting  rings,  as  shown  in  Figs. 
lOO  and  loi.  Fig.  lOO  shows  a  two-phase  concentrated  wind- 
ing, one  slot  per  pole  for  each  phase.     Fig.  loi   shows  a  two- 


FlG.    lOO. 


phase  winding  distributed  in  two  slots  per  pole  for  each  phase. 

Three-phase  windings. — The  three-phase  winding  is  three  inde- 
pendent single-phase  windings  on  the  same  armature,  the  term- 
inals of  the  individual  windings  being  connected  according  to  the 
P^scheme  or  ^/-scheme,  as  explained  in  Art.  6^.  Fig.  102  shows 
a  three-phase  concentrated  winding,  one  slot  per  pole  for  each 
phase  ;  Y  connected.  Fig.  103  shows  the  same  winding  J  con- 
nected. The  Y  connection  gives  \/3  times  as  much  e.  m.  f  be- 
tween collecting  rings  as  the  A  connection  for  the  same  winding. 
The  F  connection  is  more  suitable  for  high  e.  m.  f  machines  and 
the  d  connection  for  machines  for  large  current  output.  The 
""Ine  current  is  \/3  times  as  great  as  the  current  in  each  winding 


Fig.   I02. 


(109) 


(ho) 


Fig.  104. 


ALTERNATORS. 


Ill 


in  a  J  connected  armature.  Fig.  104  shows  a  three-phase  bar* 
winding  distributed  in  two  slots  per  pole  for  each  phase.  Fig. 
105  shows  a  three-phase  coil  winding  distributed  in  two  slots  per 
pole  for  each  phase  and  arranged  in  two  layers,  there  being  as 
many  coils  on  the  armature  as  there  are  slots,  so  that  portions  of 


Fig.  105. 


two  coils  lie  in  each  slot,  one  above  the  other.  The  portions  of 
the  coils  represented  by  full  lines  lie  in  the  upper  parts  of  the 
slots  and  the  adjacent  dotted  portions  lie  in  the  bottoms  of  the 
same  slots. 

The  Y  connection. — The  terminals  of  the  individual  windings  which  are  to  be  con- 
nected to  the  common  junction  and  to  the  collecting  rings  may  be  determined  as 
follows  :  Consider  the  instant  when  winding  A  is  squarely  under  the  pole  as  shown 
in  Fig.  102 ;  the  e.  m.  f.  in  this  winding  (and  current  also  if  the  circuit  is  non-in- 
ductive) is  a  maximum  and  the  currents  in  the  other  two  phases  B  and  C  are  half  as 

*  One  conductor  in  each  slot.  This  conductor  is  usually  in  the  form  of  a  copper 
bar  of  rectangular  cross  section. 


112       THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 

great.     If  winding  A  is  connected  so  that  its  current  is  flowing  away  from  J^,  wind- 
ings B  and  C  must  be  connected  so  that  their  currents  flow  towards  K. 

The  [\  connection. — The  three  windings  form  a  closed  circuit  when  ^  connected. 
The  total  e.  m.  f.  around  this  circuit  at  any  instant  must  be  zero.  Therefore  the  e. 
m.  f.  in  winding  A  when  it  is  directly  under  the  poles  must  oppose  the  e.  m.  f. '  s  of 
windings  B  and  C. 

85.  Insulation  of  armatures. — Armatures  for  alternators  must  be 
well  insulated  in  cases  where  the  e.  m.  f.  generated  is  high  as  the 
e.  m.  f.,  tending  to  break  down  the  insulation  is  the  maximum 
value  of  the  e.  m.  f  generated,  and  this  is  considerably  greater 
than  the  rated  or  effective  e.  m.  f.  Concentrated  or  partially  dis- 
tributed windings  admit  of  quite  a  high  degree  of  insulation  in- 
asmuch as  the  slots  may  be  made  quite  large  and  there  are 
comparatively  few  crossings  of  the  coils  at  the  ends  of  the  ar- 
mature. Distributed  windings  can  not  be  so  highly  insulated 
because  there  are  many  crossings  of  the  coils  and  the  slots  are 
necessarily  small,  such  windings  are,  therefore,  not  suitable  for 
the  generation  of  high  e.  m.  f 's.  Alternators  having  this  type 
of  winding  should  therefore  be  used  in  connection  with  step-up 
transformers  if  a  high  e.  m.  f  is  desired.  When  it  is  desired  to 
generate  a  high  pressure  directly  it  is  best  to  use  a  machine  with 
a  stationary  armature.  Such  armatures  have  been  built  for  e. 
m.  f 's  of  8,000  or  10,000  volts,  thus  doing  away  with  the  neces- 
sity of  step-up  transformers  for  power  transmission  lines  of 
moderate  length.  There  is  usually  more  room  for  thorough 
insulation  on  such  armatures  and  the  insulation  is  less  liable  to 
deteriorate  as  it  is  not  disturbed  in  any  way  by  motion  of  the 
armature.  Moreover,  the  use  of  the  stationary  armature  does 
away  with  collector  rings  and  brushes  (for  the  armature)  and  the 
consequent  necessity  of  their  insulation  for  high  potentials. 

The  individual  coils  of  an  alternator  armature  are  generally 
heavily  taped  and  treated  with  insulating  oil  or  varnish,  the  slots 
are  lined  with  heavy  tubes  built  up  of  paper  and  mica  and  all 
parts  of  the  core  which  are  near  the  coils  are  also  covered  with  a 
heavy  layer  of  insulating  material. 


6ooo  " 

5000 

5000  " 

4500 

4500  " 

4000 

4000  " 

3500 

3500  " 

3000 

ALTERNATORS.  II3 

85.  Magnetic  densities  in  armature  and  air  gap. — The  armature 
core  is  usually  made  of  sufficient  cross  section  to  insure  a  fairly- 
low  magnetic  density.  This  is  done  in  order  to  keep  down  the 
hysteresis  and  eddy  current  losses  which  would  otherwise  be  high 
on  account  of  the  comparatively  high  frequencies  employed. 
The  allowable  magnetic  density  in  the  armature  core  depends 
largely  upon  the  frequency,  since  the  density  for  a  given  loss  may 
be  higher  the  lower  the  frequency.  The  following  table  from 
Kolben  gives  values  of  the  density  suitable  for  various  frequencies. 

B.  Lines  per  cm^. 
40  cycles 6500  to  5500 

50 
60 
80 

100 

120 

The  allowable  magnetic  density  in  the  air  gap  will  depend  to  some 
extent  upon  the  material  used  for  the  pole  pieces.  With  cast-iron 
poles  this  density  should  not  exceed  4000  to  4500  lines  per  sq. 
cm. ;  with  wrought-iron  pole  pieces  it  may  be  as  high  as  6000  to 
7000  lines  per  sq.  cm. 

86.  Current  densities. — The  current  density  in  early  alternator 
armatures  was  often  very  high  ;  not  more  than  300  circular  mils 
per  ampere  being  allowed  in  many  cases.  Such  armatures  usually 
ran  very  hot  at  full  load.  The  current  densities  used  in  modern 
machines  are  much  lower,  from  500  to  700  c.  mils  per  ampere 
being  allowed,  as  in  the  case  of  direct  current  machines.  The 
armature  conductor  is  usually  of  ordinary  cotton  covered  mag- 
net wire  in  the  smaller  machines,  and  when  a  conductor  of  con- 
siderable cross  section  is  required  a  number  of  wires  are  grouped 
in  multiple.  In  larger  machines  copper  bars  are  frequently  used, 
as  these  admit  of  a  large  cross  section  being  put  in  a  minimum 
space.  Wire  of  rectangular  cross  section  and  copper  ribbon  are 
also  used  in  some  cases. 

8 


114   THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

88.  Outline  of  alternator  design. — An  alternator  is  usually  de- 
signed to  give  an  e.  m.  f.  of  prescribed  value  and  frequency,  and 
to  be  capable  of  delivering  a  prescribed  current  without  undue 
heating.  To  design  an  alternator  is  to  so  proportion  the  parts 
as  to  satisfy  the  following  conditions. 

(a)  The  product  of  revolutions  per  second  into  the  number  of 
pairs  of  field  magnet  poles  must  equal  the  prescribed  frequency 
according  to  equation  (i8). 

(b)  Equation  (59)  namely 

AA\kNTf 

^  —  s 

must  be  satisfied,  to  give  the  prescribed  e.  m.  f. 

(c)  Peripheral  speed  of  armature  must  not  exceed  allowable 
limits. 

(d)  The  armature  must  have  sufficient  surface  to  radiate  the 
total  watts  lost  in  the  armature  (including  eddy  surrent  and  hys- 
teresis losses)  without  excessive  rise  of  temperature.  There  are 
two  distinct  cases  in  the  designing  of  an  alternator  as  follows  : 

Case  I. — Where  the  speed  is  fixed  by  independent  considera- 
tions, as,  for  example,  in  direct  connected  machines.  In  this 
case  the  number  of  poles  is  determined  by  the  given  speed  and 
prescribed  frequency.  The  diameter  of  the  armature  follows 
from  the  allowable  peripheral  speed.*  Assuming  from  2%  to 
5  %  t  of  total  rated  output  as  armature  loss,  the  approximate 
length  of  the  armature  is  then  determined  by  the  radiating  sur- 
face required.  A  well  ventilated  armature  should  radiate  from 
.05  to  .06  watt  per  square  inch  (of  cylindrical  surface)  per  de- 
gree Centigrade  rise  of  temperature.  This  constant  of  radiation 
varies  greatly  with  the  style  of  construction  of  the  armature  and 
with  peripheral  speed.  The  length  may  be  slightly  modified 
when  condition  (d)  comes  to  be  considered.     The  flux  N  is  de- 

*  Direct  connected  dynamos  are  scarcely  ever  run  up  to  the  allowable  peripheral 
speed.     Speeds  from  2,200  to  2,600  feet  per  minute  are  usual. 

•j-  According  to  size  of  machine,  low  percentage  being  for  large  machines. 


ALTERNATORS.  11$ 

termined  from  the  flux  density  in  the  air  gap  and  the  area  of  each 
pole-face.  The  combined  area  of  the  pole-faces  is  usually  about 
equal  to  half  the  cylindrical  surface  of  the  armature,  or,  in  other 
words,  the  distance  between  tips  of  adjacent  poles  is  equal  to  the 
breadth  of  the  pole  face.  The  number  of  armature  turns  T  is  then 
determined  from  equation  (59).  The  armature  turns  thus  deter- 
mined may  come  out  an  odd  or  a  fractional  number  and  must  be 
adjusted  to  suit  the  type  of  winding  employed,  that  is,  to  give  the 
required  number  of  coils  each  having  the  same  number  of  turns. 
The  length  of  the  armature  may  then  be  changed  slightly  to 
adjust  N  so  that  the  required  e.  m.  f  will  be  produced  with  the 
adopted  number  of  armature  turns.  The  area  of  cross-section 
of  the  armature  conductors  is  fixed  by  the  allowable  current  den- 
sity and  rated  current  output. 

Case  IT. — When  speed  is  not  fixed  by  independent  considera- 
tions. In  this  case  a  trial  combination  of  poles  and  speed  is 
adopted  giving  a  speed  suitable  for  the  size  of  the  armature. 
The  remainder  of  the  design  is  then  worked  out  as  above. 

Remark :  When  a  machine  is  provisionally  designed  the  details 
of  its  behavior  may  be  approximately  calculated  without  difficulty, 
and  refinement  of  design  is  attained  by  working  out  a  number  of 
provisional  designs  and  calculating  the  details  of  their  action, 
then  the  most  satisfactory  design  may  be  recognized  and  adopted. 

The  proportioning  of  the  magnetic  circuits  and  the  calculation 
of  field  windings  of  an  alternator  is  carried  out  in  the  same  gen- 
eral way  as  in  the  case  of  a  direct-current  dynamo. 

Remark  :  In  designing  a  two-phase  alternator  each  winding  is 
allowed  to  cover  half  of  the  armature  surface.  In  designing  a 
three-phase  alternator  each  winding  is  allowed  to  cover  one  -third 
of  the  armature  surface. 

89.  Field  excitation  of  alternators. — The  use  of  an  auxiliary 
direct-current  dynamo  for  exciting  the  field  of  an  alternator  has 
been  pointed  out  in  Art.  19.  The  e.  m.  f  of  an  alternator  ex- 
cited in  this  Avay,  falls  off  greatly  with  increasing  current  output 


Il6      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

and  to  counteract  this  tendency  an  auxiliary  field  excitation  is 
frequently  provided  which  increases  with  the  current  output  of 
the  machine.  For  this  purpose  the  whole  or  a  portion  of  the 
current  given  out  by  the  machine  is  rectified  *  and  sent  through 
the  auxiliary  field  coils. 

Fig.  1 06  shows  an  alternator  A  with  its  field  coils  /^separately 


Fig.  106. 


excited  from  a  direct-current  dynamo  E.  The  two  rheostats  R 
and  r,  in  series  with  the  alternator  and  exciter  fields  respectively, 
are  used  to  regulate  the  field  current.  Fig.  107  shows  an  alter- 
nator with  two  sets  of  field  coils  F  and  C.  The  coils  F  are  sep- 
arately excited  as  before.  The  coils  C,  known  as  the  series  or 
compound  coils,  are  excited  by  current  from  the  armature  of  the 
alternator.  One  terminal  of  the  armature  winding  is  connected 
directly  to  a  collecting  ring.  The  other  armature  terminal  con- 
nects to  one  set  of  bars  in  the  rectifying  commutator  B.  From 
the  rectifier  the  current  is  led  through  the  winding  C,  thence 
back  to  the  rectifier  and  thence  to  the  second  collecting  ring. 
The  rectifying  commutator  B  is  provided  with  as  many  segments 

*  Connections  to  field  coils  are  reversed  with  every  reversal  of  main  current  so  that, 
in  the  field  coils,  the  current  is  unidirectional. 


ALTERNATORS. 


117 


as  there  are  poles  on  the  machine.  This  commutator  re- 
verses the  connections  of  the  terminals  of  the  coils  C  at  every 
pulsation  of  the  alternating  current  so  that  the  current  flows 
in  C  always  in  the  same  direction.  The  commutator  B  is 
fixed  to  the  armature  shaft.  A  shunt  s  moving  with  the  com- 
mutator is  sometimes  used  when  it  is  desired  to  rectify  only  a 
portion  of  the  current.  A  stationary  shunt  s'  is  also  frequently 
used  to  regulate  the  amount  of  current  flowing  around  the  coils 
Cy  thus  giving  a  method  of  adjusting  the  compounding. 


Fig.  107. 

Fig.  108  shows  an  alternator  A  with  two  sets  of  field  coils  F 
and  C  as  before.  One  armature  terminal  is  connected  to  a  col- 
lecting ring  and  the  other  armature  terminal  connects  to  the  pri- 
mary of  a  transformer  T  and  thence  to  the  other  collecting  ring. 
The  terminals  of  the  secondary  coil  of  T  connect  to  the  bars  of 
the  rectifying  commutator  B  from  which  the  compound  field 
winding  C  is  supplied.  The  transformer  T  is  usually  placed  in- 
side the  armature.    All  three  of  the  methods  shown  in  Figs.  106, 


Il8      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

107  and  108  are  in  common  use  for  the  field  excitation  of  alter- 
nators.    Compounding  is  necessary  only  with  alternators  which 


have  fairly  high  armature  inductance  and  which,  with  constant 
field  excitation,  would  give  poor  regulation.  For  low  inductance 
machines  the  separate  excitation  alone  is  usually  sufficient. 


CHAPTER  X. 

THE  TRANSFORMER. 

90.  The  transformer  consists  of  an  iron  core  upon  which  two 
separate  and  distinct  coils  of  wire  are  wound.  When  one  of 
these  coils  receives  alternating  current  from  any  source  the  other 
coil  delivers  alternating  current  to  any  circuit  which  may  be  con- 
nected to  its  terminals.  The  coil  which  receives  current  is  called 
XhQj)rimary  and  the  coil  which  delivers  current  is  called  the  sec- 
ondary.  The  alternating  current  in  the  primary  coil  magnetizes 
and  demagnetizes  the  iron  core  repeatedly  thus  inducing  an  alter- 
nating e.  m.  f.  in  the  secondary  coil. 

For  the  purpose  of  the  immediately  following  discussion  the 
resistances  R'  and  R"  of  the  respective  coils  are  supposed  to  be 
negligibly  small ;  the  magnetic  reluctance  of  the  core  is  supposed 
to  be  negligibly  small  so  that  a  very  weak  magnetomotive  force 
may  produce  a  very  large  flux  through  the  core ;  and  all  the 
magnetic  flux  TV  which  passes  through  the  primary  coil  is  sup- 
posed to  pass  through  the  secondary  coil  also.  A  transformer 
satisfying  these  conditions  would  be  called  an,  ideal  transformer. 

91.  Ratio  of  transformation. — Let  an  alternating  e.  m.  f.  (not 

necessarily  harmonic)  of  which  the  instantaneous  value  is  e'  act 

on  the  primary  coil  of  a  transformer.     Since  R'  is  negligible  this 

dN 
e.  m.  f  is  all  balanced  by  the  counter  e.  m.  f.,  Z'  — i-,  which  is 

induced  in  the  primary  coil  by  the  changing  core  flux  N.    Therefore 

.  =  Z'f  ■  (60) 

in  which  Z'  is  the  number  of  turns  of  wire  in  the  primary  coil. 
The  changing  flux  induces  in  the  secondary  coil  an  alternating 
e.  m.  f.  of  which  the  instantaneous  value  is  : 

(119) 


I20      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

^'=-^"'f  (61) 

in  which  Z"  is  the  number  of  turns  of  wire  in  the  secondary  coil. 
This  e.  m.  f.  e"  is  all  available  at  the  terminals  of  the  secondary 
coil  since  R'  is  negligible.     Equations  (60)  and  (61)  show  that 

Z" 

e"  is  at  each  instant  equal  to  -yje'  so  that 

E'       Z' 

in  which  E'  and  E"  are  the  effective  primary  and  secondary  e. 
m.  f.'s  respectively. 

When  the  circuit  of  the  secondary  coil  is  open  the  current  in 
it  is  zero  and  the  only  current  in  the  primary  coil  is  the  negligible 
current  required  to  magnetize  the  core.  When  the  secondary 
circuit  is  closed  a  current  flows  in  it.  Let  i"  be  the  instantaneous 
value  of  this  current  then  the  instantaneous  value  i'  of  the  pri- 
mary current  is  siich  that 

Z'i'  +  Z"i"  =  o.  (63) 

That  is,  the  primary  ampere  turns,  Z'i' ,  are  equal  and  oppo- 
site to  the  secondary  ampere  turns  Z"i"  (reluctance  of  core  being 

i'  Z" 

negligible).     Equation  (63)  shows  that  ^  is  always  equal  to  -^— . 

Therefore 

/'       Z" 

Y>  =  z~  <^^4) 

in  which  /'  and  I"  are  the  effective  values  of  primary  and  sec- 
ondary currents  respectively. 

In  an  ideal  transformer,  then,  the  secondary  e.  m.  f.  E"  is  oppo- 

Z" 

site  to  the  primary  e.  m.  f.  E'  in  phase  and  -™   times  as  great ; 

and  the  primary  current  I'  is  opposite  to  the  secondary  current 

Z" 
I"  in  phase  and  ™-  times  as  great.     Therefore  the  angular  phase 

difference  d  between  secondary  e.  m.  f.  and  secondary  current  is 


THE  TRANSFORMER. 


121 


equal  to  the  angular  phase  difference  between  primary  e.  m.  f. 
and  primary  current ;  and  the  power  intake  of  the  primary 
E' I'  cos  d,  is  equal  to  the  power  output  of  the  secondary 
E"I"  cos  Q. 

Z" 

The  ratio  ^is  called  the  ratio  of  transformation  of  the  trans- 
former. 

92.  Particular  cases  i^for  harmonic  e.  m.  f.  and  curreni).  i. 
Non-inductive  receiving  circuit. — In  this  case  E"  and  I"  are  in 
phase  and  therefore  E  and  I'  are  in  phase  also.  The  state  of 
affairs  is  represented  in  Fig.  109.     The  line  ON  represents  the 


I' 


-y 


^E" 


•N 


Fig.  109. 

harmonically  varying  core  flux,  OE'  represents  the  e.  m.  f  acting 
on  the  primary,  and  OE"  represents  the  e.  m.  f  induced  in  the 
secondary. 

2.  Inductive  secondary  receiving  circuit. — In  this  case  I"  lags 

behind  E"  by  the  angle  whose  tangent  is  —^ ,  where  Z.*  is  the  in- 


*  The  inductance  of  the  secondary  coil  itself  is  already  accounted  for  in  the  action 
of  the  transformer. 


122       THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 

ductance  and  R  the  resistance  of  the  receiving  curcuit.  Also  /' 
lags  behind  E'  by  the  same  angle.  The  state  of  affairs  is  shown 
in  Fig.  no. 

3.  Secondary    receiving    circuit   containing    a    condenser. — In 
this  case  /"  is   ahead   of  E"  by  the  angle  whose  tangent  is 

I- — -J—  ojL  \  -i-  R,  where y  is  the  capacity  of  the  condenser,  L  is 

the  inductance  of  the  connecting  wires  and  R  their  resistance. 


^^J?^^^' 


*-JV 


Fig.  112. 


Also,  /'  is  ahead  of  E'  by  the  same  angle.     The  state  of  affairs 
IS  shown  in  Fig.  in. 

93.  Equivalent  resistance  and  reactance  of  an  ideal  transformer 
feeding  a  given  secondary  circuit. — ^The  primary  of  a  transformer 
takes  from  the  mains  a  definite  current  at  a  definite  phase  lag 
when  the  secondary  is  supplying  current  to  a  given  circuit. 
Consider  a  simple  circuit  of  resistance  r  and  reactance  x  which, 
connected  to  the  mains,  takes  the  same  current  as  the  primary 
of  the  transformer  and  at  the  same  phase  lag.  This  circuit  is 
said  to  be  equivalent  to  the  transformer  and  its  secondary  receiv- 


THE   TRANSFORMER.  1 23 

ing  circuit,  and  r  and  x  are  called  the  equivalent  primary  resis- 
tance and  reactance  respectively  of  the  secondary  receiving  circuit. 
Resolve  the  primary  e.  m.  f.  E ,  Fig.  112,  into  components 
parallel  to  /'  and  perpendicular  to  /'  as  shown.  The  compo- 
nent parallel  to  I'  vs,  rl'  and  the  component  perpendicular  to  /' 
is  xl'  according  to  problem  IV.  The'  triangle  whose  sides  are 
E ,  rl'  and  xl'  is  similar  to  the  triangle  whose  sides  are  E' , 
RI"  and  XI" ,  R  and  X  being  the  given  resistance  and  reactance 
of  the  secondary  receiving  circuit.     Therefore 

xP  _E 
~XT''~~E' 

rP   _E 
W"  ^  ~E'' 


and 


E      Z'  I'      Z" 

But  -^,  =  yj,  and  -jj,  =  y,  so  that 


\  (65) 

X.  (66) 


con- 


That  is,  a  transformer  supplying  current  from  its  secondary  to 
a  circuit  of  resistance  R  and  reactance  X,  takes  from  the  mains 
the  same  current  at  the  same  phase  lag  as  would  be  taken  by  a 

(Z'  \^  I Z'  \^ 

-yYi  I  R  and   of  reactance  I  -yj,  1  X 

nected  directly  to  the  mains. 

94.  Maximum  core  flux. — When  the  e.  m.  f  acting  on  the 
primary  of  a  transformer  is  harmonic  there  is  a  simple  and  im- 
portant relation  between  E ,  o),  Z  and  the  maximum  core  flux 
JV".  Let  e'  be  the  in.stantaneous  value  of  fhe  primary  e.  m.  f 
Then  since  e^  is  harmonic  we  have 

c'  =  'E'  sin  (Jit  (67) 

in  which  ^'  is  themaximum  value  of  ^'.     Substituting  the  value 
of  e'  from  {6']')  in  (60)  we  have 


124      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 
dN      ^'     . 

-di=  zr  ^^^  "^ 
whence  by  integration : 

N= yy  COS  a)t.  (68) 

That  is,  the  core  flux  iV  is  a  harmonically  varying  quantity  and 
its  maximum  value  is  : 


N= 


coZ' 


or  N=^^^\  (69) 

95.  The  actual  transformer. — The  action  of  the  actual  trans- 
former deviates  from  the  above  described  ideal  action  on  account 
of  the  resistances  R'  and  R"  of  the  coils ;  on  account  of  eddy 
currents  and  hysteresis ;  and  on  account  of  the  fact  that  some 
lines  of  flux  pass  through  one  coil  without  passing  through  the 
other  (magnetic  leakage). 

However,  a  well  designed  transformer  at  moderate  load  ap- 
proximates so  closely  to  the  ideal  transformer  in  its  action  that 
equations  (62)  to  (69)  may  be  used  in  practical  calculations. 

In  the  following  discussion  the  effects  of  core  reluctance  and 
eddy  currents,  of  coil  resistances,  and  of  magnetic  leakage  are 
considered  separately.  These  effects  are  small  in  themselves  and 
their  influence  on  each  other  is  entirely  negligible. 

96.  Effects  of  core  reluctance  and  eddy  currents  upon  the  action 
of  a  transformer. — The  magnetizing  action  of  the  primary  cur- 
rent on  the  core  of  a  transformer  exceeds  the  opposite  magnetiz- 
ing action  of  the  secondary  current  by  an  amount  sufficient  to 
balance  the  demagnetizing  action  of  the  eddy  currents  in  the  core 
and  to  overcome  the  magnetic  reluctance  of  the  core.  Therefore, 
the  primary  current  may  be  considered  in  three  parts,  namely : 
the  part  a  which  counteracts  the  magnetizing  action  of  the  sec- 
ondary current ;  the  part  m  the  magnetizing  action  of  which  over- 


THE  TRANSFORMER. 


125 


comes  the  reluctance  of  the  core ;  and  the  part  c  which  counter- 
acts the  magnetizing  action  of  the  eddy  currents.  Thus  OA,  Fig. 
113,  represents  the  part  a  of  the  primary  current,  this  part  is  op- 

Z" 
posite  to  /''  in  phase  and  equal  to  y^  xl";  OM  represents  the 

part  m ;  and  OC  represents  the  part  c.  The  line  01'  which  is 
the  vector  sum  of  OA,  OM  and 
OC  represents  the  total  primary 
current.  The  currents  ui  and  c  are 
usually  small  compared  with  the 
full  load  primary  current  of  a 
transformer.  The  currents  m  and 
c  together,  namely  111  +  c,  consti- 
tute what  is  called  the  leakage  or 
magnetizing  current  of  the  trans- 
former. 

Discussion  of  the  part  c  of  the 
primary  current. — This  part  of  the 
primary  current  is  "harmonic  when 
the  primary  e.  m.  f  is  harmonic 
and  it  is  in  phase  with  the  pri- 
mary e.  m.  f*  Let  C  be  the 
effective  value  of  the  current  c ;  then  the  power  taken  from 
the  mains  by  the  current  c\\s  E  C  and  this  power  is  equal  to  the 
eddy  current  loss,  W^,  in  the  core,     Therefore 

W 
E' 


'2f 


Fig.  113. 


c= 


(70) 


The  method  of  calculating  eddy  current  loss  W^  is  given  later. 
Discussion  of  the  par^t  m  of  the  pidmary  current. — This  part  of 

*  Proof:  The  primary  e.  m.  f.  and  the  eddy  currents  are  at  their  maximum  value 
at  the  instant  when  the  core  flux  is  changing  fastest.  That  is  eddy  currents  are  in 
phase  with  primary  e.  m.  f.  (really  in  opposition  to  primary  e.  m.  f. ).  The  current  c 
which  opposes  the  magnetizing  action  of  the  eddy  currents  is  of  course  opposite  to 
eddy  cuiTcnts  in  phase  and  therefore  in  phase  with  primary  e.  m.  f. 


126      THE    ELEMENTS   OF   ALTERNATING   CURRENTS. 

the  primary  current  is  not  harmonic,  but  since  it  is  small  it  may 
be  treated  as  a  harmonic  current  without  great  error.  Let  M  be 
the  effective  value  of  the  current  in ;  let  M'  be  the  component  of 
TJ/ parallel  to  E  \  and  let  M"  be  the  component  of  M  perpen- 
dicular to  E' .  Then  E  M'  is  the  power  taken  from  the  mains 
by  the  current  vi  and  this  power  is  equal  to  the  hysteresis  loss, 
W^,  in  the  core.     Therefore 

W 
M  =  ^^  (a) 

The  component  M"  of  the  current  m  reaches  its  maximum 
value  s/  2M"  when  the  core  flux  is  at  its  maximum  value  N  and 
inasmuch  as  this  component  of  in  overcomes  the  reluctance  of 
the  core  we  have 

47Z' 

M"  =  ^-^  (b) 

Af\^  27tZ' 

in  which  G  is  the  magnetic  reluctance  of  the  core. 

Total  leakage  current. — From  equations  (70),  (a)  and  (b)  we 

have 

W  ^  W 
Power  component  of  leakage  current  =  — ^-^^7 — ^       (71) 

r    ,        ^  lONG  ^        . 

Wattless  component  of  leakage  current  = = (72) 

4\^2TrZ' 

in  which  G  is  the  magnetic  reluctance  of  the  core,  Z'  is  turns  of 

wire  in  primary  coil,  and  JV  is  the  maximum  value  of  the  core  flux. 

97.  Actual  value  of  the  part  m  of  the  primary  current. 

(a)  IVheti  the  co7'e  is  assumed  to  be  without  hysteresis. — Let  the  ordinates  ot  the 
curve  7nN,  Fig.  114,  represent  values  of  core  flux  N  produced  by  various  given  cur- 
rent strengths  m  in  the  primary  coil,  these  current  strengths  being  represented  by  the 
abscissas  of  the  curve  niN. 

When  the  primary  e.  m.  f.  is  harmonic  then  the  core  flux  N  is  harmonic  also,  and 
90°  behind  E'  in  phase,  according  to  equation  (68).  Let  the  sine  curve  Ny  Fig.  I14, 
represent  the  value  of  A' as  time  passes  ;  time  as  abscissas  and  A'' as  ordinates. 


THE   TRANSFORMER. 


127 


Then  the  curve  m  of  which  the  ordinates  represent  successive  instantaneous  values 
of  the  current  m  is  constructed  as  follows  :  Draw  the  ordinate  dp  and  the  abscissa  ap. 
Lay  of  dc  equal  to  ab  which  is  the  magnetizing  cvurent  required  to  force  through  the 
core  the  flux  dp.  The  locus  of  the  point  c  is  the  required  curve.  The  figure  shows 
that  the  magnetizing  current  is  not  harmonic  although  it  is  wattless. 


1 

vnN 

— "v^ 

\ 

l/J^ 

\\ 

i/^^'d. 

^A 

axis  of  iimc  and  on          .^ 

0 

\ 

\^                              y/N 

Fig.  114. 

(^)  When  the  hysteresis  is  taken  into  account. — Let  the  ordinates  of  the  curve  mN^ 
Fig.  115,  represent  values  of  core  flux  produced  by  various  given  current  strengths 
in  the  primary  coil,  these  current  strengths  being  represented  by  the  abscissas  of  the 
curve  mN.    The  curve  m.  of  which  the  ordinates  represent  the  successive  instantaneous 


Fig.  115. 


values  of  the  current  m  is  constructed  as  before  ;  the  ascending  branch  of  the  hysteresis 
loup  mN  being  used  for  increasing  values  of  N  and  the  descending  branch  for  decreas- 
ing values  of  N. 

98.  Transformer   regulation.     Preliminary  statement  concern- 
ing the  effects  of  magnetic  leakage  and  of  resistances  of  primary 


128   THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

and  secondary  coils  on  the  action  of  a  transformer. — In  the  ideal 
transformer  the  whole  of  the  primary  e.  m.  f.  is  balanced  by  the 
opposite  e.  m.  f.  induced  in  the  primary  coil  by  the  varying  mag- 
netic flux  which  passes  through  both  coils,  and  the  whole  of  the 

e.  m.  f.  induced  in  the  secondary  coil  by  this  varying  flux  is 
available  at  the  terminals  of  the  secondary  coil. 

In  the  actual  transformer  a  portion  of  the  primary  e,  m.  f.  is 
lost  in  overcoming  the  resistance  of  the  primary  coil  and  a  por- 
tion is  lost  in  balancing  the  e.  m.  f.  which  is  induced  in  the  pri- 
mary coil  by  the  flux  which  passes  through  the  primary  coil 
but  does  not  pass  through  the  secondary  coil  (leakage  flux). 
These  lost  portions  of  the  primary  e.  m.  f  are  proportional  to 
the  primary  current  so  that  the  useful  part  *  of  the  primary  e.  m. 

f.  falls  short  f  of  the  total  primary  e.  m.  f.  by  an  amount  which  is 
proportional  to  the  current. 

The  total  e.  m.  f.  induced  in  the  secondary  coil  is  proportional 
to  the  usefd  part  of  the  primary  e.  m.  f.  and  a  portion  of  the 
total  secondary  e.  m.  f.  is  lost  in  overcoming  the  resistance  of 
the  secondary  coil.  This  lost  portion  of  the  secondary  e.  m.  f. 
is  proportional  to  the  secondary  current  (or  to  primary  current, 
since  the  ratio  of  the  currents  is  constant). 

Therefore  the  eflect  of  magnetic  leakage  and  of  coil  resistances 

is  to  make  the  e.  m.  f  between  the  terminals  of  the  secondary 

Z" 
coil  fall  short  f  of  its  ideal  value  ^7-  ■  E'  by  an  amount  which  is 

proportional  to  the  current. 

This  falling  off  of  secondary  e.  m.  f.  with  increasing  current  is 
of  practical  importance  inasmuch  as  most  receiving  apparatus 
must  be  supplied  with  current  at  approximately  constant  e.  m.  f. 
A  transformer  of  which  the  secondary  e.  m.  f.  falls  off  but  little 


*  The  part,  namely,  which  balances  the  e.  m.  f.  induced  in  the  primary  coil  by  the 
magnetic  flux  which  passes  through  both  coils. 

t  The  lost  portions  of  primary  and  secondary  e.  m.  f.'s  are  in  general  not  m  phase 
with  total  primary  and  total  secondary  e.  m.  f.'s.  These  losses  are  therefore  to  be 
subtracted  as  vectors  as  explained  in  the  following  articles. 


THE   TRANSFORMER. 


129 


with  increase  of  current  is  said  to  have  good  regulation.  A 
transformer  to  regulate  well  must  have  low  resistance  coils  and 
little  magnetic  leakage.  Large  transformers  as  a  rule  regulate 
more  closely  than  small  ones. 

99.  Effect  of  resistance  of  coils  upon  the  action  of  a  trans- 
former.— Fig.  116  shows  the  general  effect  of  the  resistances  of 
the  coils  upon  the  action  of  a  transformer.  The  line  ON  repre- 
sents the  harmonically  varying  flux  in  the  core.  Oa  represents 
the  useful  part  of  the  primary 
e.  m,  f.  and  Ob  the  total  e.  m.  f. 
induced  in  the  secondary  coil. 
The  line  01"  represents  the  sec- 
ondary current  and  the  line  01' 
represents  the  primary  current. 
The  total  primary  e.  m.  f.  E' 
exceeds  Oa  by  the  amount  R'  P 
(parallel  to  /^),  and  the  e.  m.  f. 
E"  at  the  terminals  of  the  sec- 
ondary coil  falls  short  of  Ob  by 
the  amount  R"I"  (parallel  to  I"). 

Remark :  When  the  angle  d, 
Fig.  116,  is  nearly  zero  (second- 
ary receiving  circuit  noninduc- 
tive)  then    R' F    and   R'T'  are 

nearly  parallel  to  Oa  and  Ob  respectively,  so  that  Oa  is  much 
less  than  E'  in  value  and  E"  is  much  less  than  Ob  in  value.  On 
the  other  hand,  when  the  angle  Q  is  nearly  ±90°  (secondary  re- 
ceiving circuit  containing  a  large  inductance  or  a  condenser) 
then  R' I'  and  RI'I"  are  nearly  perpendicular  to  Oa  and  Ob  re- 
spectively, so  that  Oa  is  nearly  equal  to  E'  in  value  and  E"  is 
nearly  equal  to  Ob  in  value.  Therefore  the  regulation  of  a  trans- 
former is  largely  affected  by  coil  resistance  when  the  secondary 
receiving  circuit  is  noninductive,  but  scarcely  at  all  affected  by 

9 


-^jy 


Fig  116. 


I30      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

the  coil  resistance  when  the  secondary  receiving  circuit  contains 
a  large  inductance  or  a  condenser. 

100.  Effect  of  magnetic  leakage  upon  the  action  of  a  trans- 
former.— It  is  shown  in  the  next  article  that  magnetic  leakage  is 
in  its  effects  equivalent  to  an  auxiliary  outside  inductance  P 
through  which  the  primary  current  passes  on  its  way  to  the  pri- 
mary of  the  transformer.  The  part  of  the  primary  e.  m.  f.  E' 
which  is  lost  in  this  inductance  is  equal  to  oiPI'  and  it  is  90° 
ahead  of  the  primary  current  I'  in  phase. 


^N 


Fig.  117. 

When  the  secondary  receiving  circuit  is  inductive  I"  lags  be- 
hind E"  {Oh,  Fig.  1 17)  by  the  angle  d  as  shown  in  Fig.  117,  and 
the  useful  part,  Oa,  of  the  primary  e.  m.  f  is  less  than  the  total 
primary  e.  m.  f.     In  this  case  the  secondary  e.  m.  f.,  which  is  equal 

Z" 

to  -^  X  Oa,  falls  off  in  value  as  /'  (and  also  (oPI'')  increases. 

When  the  secondary  receiving  circuit  contains  a  condenser  I" 
is  ahead  of  E"  {Ob,  Fig.  1 1 8)  as  shown  in  Fig.  1 1 8,  and  the  useful 
part,  Oa,  of  the  primary  e.  m.  f.  is  greater  than  the  total  primary 
e.  m.  f  in  value.     In  this  case  the  secondary  e.  m.  f.,  which  is  equal 

to  ^=^  X  Oa,  increases  in  value  as  /'  (and  also  coPI')  increases. 


THE   TRANSFORMER. 


131 


When  the  secondary  receiving  circuit  is  noninductive  the  angle 
d  is  zero  and  wPI'  is  at  right  angles  to  Oa  so  that  Oa  is  sensibly 
equal  to  E'  in  value  and  therefore  sensibly  constant.  In  this  case 
the  secondary  e.  m,  f.  remains  sensibly  constant  as  /'  (and  also 
o)PI')  increases. 

101.  Proposition.  The  effect  of  magnetic  leakage  in  a  trans- 
former is  equivalent  to  a  certain  outside  inductance  P,  connected  in 
series  with  the  primary  coil. 

Discussion. — Let  A,  Fig.  119,  be  the  primary  coil,  B  the  sec- 
ondary coil,  and  C  the  iron  core  of  a  transformer.     As  the  (har- 
monic) alternating  currents 
in  A  and  B  pulsate,  har- 
monically  varying   fluxes        ^ 
are  produced  through  the 
core  and  around  the  coils.      — - 
Let  OC,  Fig.   120,  repre- 
sent the.harmonically  vary- 
ing flux  through  the  core  ;        _ 


C~ 


-*■   c 


p 


Fig.  119. 


Op  the  harmonically  vary- 
ing flux  which  encircles 
coil  A  only;  and  Os  the 
harmonically  varying  flux 
which  encircles  coil^  only. 
The  fluxes  Op  and  Os  are  proportional  to  and  in  phase  with  /'  and 
/"  respectively,  so  that  the  total  flux  Op  +  Os  (represented  by  the 

lines  sp  or  ba,  Fig.  1 20)  which 
passes  between  A  and  B  is  pro- 
portional to  and  in  phase  with  /'. 
The  total  harmonically  varying 
flux  through  coil  A  is  0C+  Op 
or  Oa,a.nd  the  total  harmonically 
varying  flux  through  coil  B  is 
OC-j-  Os  or  Od.  Now,  Oa  = 
Fig.  120.  Od  -]-  da,  so  that  we  may  look 


132      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

upon  the  action  of  the  transformer  as  due  to  the  flux  Ob  passing 
through  both  coils  and  the  flux  ba  passing  through  the  primary- 
coil  only.  This  latter  flux  being  proportional  to  the  primary  cur- 
rent is  equivalent  in  its  effects  to  an  inductance  P,  connected 
in  series  with  the  primary  coil.  Let  N'  be  the  value  of  the  leak- 
age flux  ab  which,  for  a  given  value  i'  of  the  primary  current, 
encircles  the  primary  coil,  then,  according  to  equation  (5)  we  have 

102.  Calculation  of  leakage  inductance  P. — ^The  leakage  flux 
N'  equation  (j-^)  [=  ba,  Fig.  120],  and  therefore  the  value  of 
P  also,  depends  upon  the  size  and  shape  of  the  primary  and  sec- 
ondary coils  and  upon  their  proximity  to  the  core.  In  consider- 
ing the  flux  between  the  coils  (leakage  flux)  we  need  not  consider 
whether  a  given  portion  of  this  flux  is  a  part  of  Op  or  a  part  of 
Os,  Fig.  120,  inasmuch  as  these  two  fluxes  are  added  together 
to  give  ba  ox  N' . 


Fig.  121. 


Figs.  121  and  122  show  side  and  end  views  of  a  shell  type 
transformer.  The  trend  of  the  leakage  flux  is  shown  in  the  up- 
per part  of  Fig.  122  (omitted  from  lower  part  for  the  sake  of 
clearness),  and  the  dimensions  X,  Y,  g,  I  and  /  are  shown.     Fig. 


THE   TRANSFORMER. 


133 


123  is  an  enlargement  of  the  upper  part  of  Fig.  122.     Consider 
the  flux  across  between  the  dotted  Hnes  aa,    Fig.    123.     The 

X 

magnetomotive  force  pushing  this  flux  across  is  a^kV  -p^'.*  The 

length  of  the  air  portion  of  the  magnetic  circuit  through  which 
the  leakage  flux  flows  is  /  and  its  sec- 
tional area  is  Mx  (counting  both 
limbs  of  the  coils).  Therefore,  the 
magnetic  reluctance  of  this  leakage 
.       / 


circuit  is  -^ — 1-  and    the    flux    across 
Lax 

.  .    in.  in.  f.  .      Ixdx 

between  aa  is =  ajzi'Z'   ,.,  . 

in.  r.  IX 

This  flux  encircles  the  fractional 

X 

part  -T5-  of    the    primary    turns    and, 

X 

therefore,  the  fractional  part  -r>  of  the 

flux  is  to  be  counted  as  encircling  the  entire  primary  coil  so  that 


'■m 


lea  hire 


tron  ^^ 


Isecowcfe 


1 — — 

ieco-ndai^ 

Fig.  122. 


dN'  = 


47rZ'i'X 


■  x'^dx. 


.» :         --. 


leaf[ass.  flu/ 
Iron 


T\ 


prcmarj/ 
—-% 


a 


^econdarjf 

■y— 


>5i;^^ 


Fig.  123, 


The  part  of  N'  which  flows  across  the  primary  coil  is  the  in- 
tegral of  this  expression  from  x  =  o  to  x  =  X.  This  part  of  N' 
is  therefore 


*A11  quantities  in  this  article  are  expressed  in  c.  g.  s.  units. 


134      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

4.7tZ'iaX 

Similarly,  the  part  of  N'  which  flows  across  the  secondary  coil 

is 

^Z'i'W 

The  flux  across  the  gap  g-  between  the  primary  and  secondary 
coils  is  all  counted  as  a  part  of  N'  and  is  equal  to 

^TiZ'i'Xg 
J 
Therefore 

^'=^7- [-  +  -+4  (74) 

There  is  some  leakage  flux  passing  between  the  primary  and 
secondary  coils  where  they  project  beyond  the  iron  core.  This 
part  of  the  leakage  flux  has  a  longer  air  path  than  the  leakage 
flux  which  flows  from  iron  to  iron,  say  three  times  as  long. 
Therefore,  for  X  we  may  take  the  total  length  of  the  coils  lessened 
by  say  ^  the  length  which  is  surrounded  by  air  only. 

Substituting  the  value  of  N'  from  (74)  in  (73)  we  have 

This  equation  gives  the  value  of  P  in  centimeters,  all  dimen- 
sions being  expressed  in  centimeters. 

The  equivalent  inductance  P  may  be  reduced  in  value  by  les- 
sening 1,  X,  Y  or  g  or  by  increasing  /.  Fig.  124  shows  the 
proportions  of  a  recent  type  of  transformer  for  which  the  leakage 
inductance  Pis  very  small.  The  value  of  P  may  be  further  re- 
duced by  winding  the  primary  and  secondary  coils  in  alternate 
sections. 

103.  Calculation  of  transformer  regulation. — Let  P  be  the  in- 
ductance equivalent  of  magnetic  leakage  of  a  transformer  and  let 

*  In  this  expression  Z^i^  being  equal  to  Z^^i'^  is  written  therefor. 


THE   TRANSFORMER. 


135 


R"  =  R'  ^ 


§) 


R 


where  R"  is  the  primary  resistance  which  is  equivalent  in  its 
effects  to  the  resistance  R  of  the  primary  coil  and  the  resistance 

R'  of  the  secondary  coil 
combined.  Let  x  and  r 
be  the  primary  equivalents 
of  the  reactance  X  and 
resistance  R  respectively  of 
the  secondary  receiving  cir- 
cuit. [See  equations  (65) 
and  (66).]  Then,  aside  from 
the  negligible  effects  of 
hysteresis  and  eddy  cur- 
rents on  regulation,  the 
transformer  and  its  receiv- 
ing   circuit   are    equivalent 

main 


i^ 


E' 


main 


Fig.  124. 


Fig.  125. 


to  the  circuit  shown  in  Fig.  125.  The  problem  of  transformer 
regulation  is  thus  reduced  to  the  problem  of  two  coils  in  series. 
See  Problem  VII.,  Chapter  VII. 

104.  The  constant  current  transformer. — A  transformer  of  which 
the  leakage  inductance  P  is  very  large  is  sometimes  called  a  con- 
stant current  transformer  for  the  reason  that  the  current  delivered 
by  such  a  transformer  varies  but  little  with  the  resistance  of  the 


136      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


c 

ivny^ 

\::s:n 

-v:.S-.JI 

l-.i,-:\ 

■-■^^--1 

^mm 

receiving  circuit,  so  long  as  this  resistance  is  comparatively  small, 
the  primary  of  the  transformer  being  connected  to  constant  e.  m. 

f.  mains.  The  action  of  the  in- 
ductance P  in  controlling  the  cur- 
rent is  explained  in  the  article  on 
the  constant  current  alternator. 
(Art.  78.) 

Fig.  126  is  a  sketch  of  the  Gen- 
eral   Electric    Company's    type    of 
^^°-  ^^^-  constant  current  transformer;   Cis 

the  iron  core,  PP  the  primary  coil  and  55  the  secondary  coil. 
The  secondary  coil  is  movable  and  nearly  counterbalanced,  and 
the  increased  repulsion  between  PP  and  55  due  to  a  slight  in- 
crease of  current  lifts  the  secondary  coil  to  S'S'. 

When  the  primary  and  secondary  coils  are  near  together  the 
leakage  inductance  is  very  small  and  a  decrease  in  the  resistance 
of  the  receiving  circuit  would  be  accompanied  by  a  great  increase 
of  current  were  it  not  for  the  movement  of  the  secondary  coil 
and  the  consequent  increase  of  leakage  inductance. 


CHAPTER   XI. 

TRANSFORMERS. 
{Continued. ) 

105.  Transformer  losses. — The  power  output  of  a  transformer 
is  less  than  its  power  intake  because  of  the  losses  in  the  trans- 
former. These  losses  are  :  {a)  The  iron  or  core  losses  due  to 
eddy  currents  and  hysteresis ;  and  {b)  The  copper  losses  due  to 
the  resistances  of  the  primary  and  secondary  coils. 

The  h'on  losses  are  practically  tlie  same  in  amount  at  all  loads, 
and  they  depend  upon  the  frequency  and  range  of  the  flux  den- 
sity B,  upon  the  quality  and  volume  of  the  iron,  and  upon  the 
thickness  of  the  laminations. 

The  hysteresis  loss  in  watts  is 

W^  =  aVfB'\  {y6) 

Where  /  is  the  frequency  in  cycles  per  second,  B  is  the  max- 
imum flux  density  in  lines  per  square  centimeter,  V\s,  the  volume 
of  the  iron  in  cubic  centimeters,  and  «  is  a  constant  depending 
upon  the  quality  of  the  iron.  For  annealed  refined  wrought 
iron  the  value  of  a  is  about  3  x  io~^". 
The  eddy  current  loss  in  watts  is  : 

W^=bVfH^B\  {yj) 

Where  /  is  the  thickness  of  the  laminations  in  centimeters,  and 
^  is  a  constant  depending  upon  the  quality*  of  the  iron.  For 
ordinary  iron  the  value  of  <5  is  about  1.6  x  10  ~".  Insufficient 
insulation  of  laminations  causes  excessive  eddy  current  loss. 

Remark :  Equations  {y6^  and  {jj')  may  be  used  for  calculating 
the  hysteresis  and  eddy  current  losses  in  any  mass  of  laminated 
iron  subjected  to  periodic  magnetization,  such  as  alternator  arma- 
tures and  the  rotor  and  stator  iron  in  an  induction  motor. 

*  Upon  the  specific  resistance  of  the  iron. 

(137) 


138       THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 

The  copper  loss  is  : 

W^^R'I'^^R!'I"\  (78) 

This  loss  is  nearly  zero  when  the  transformer  is  not  loaded  ;  it 
increases  with  the  square  of  the  current,  and  becomes  excessive 
when  the  transformer  is  greatly  overloaded. 

106.  Efficiency  of  transformers. — The  ratio  power  output  -~ 
power  intake  is  called  the  efficiency  of  a  transformer.  The  ac- 
companying table  shows  the  full-load  efficiencies  of  various  sized 
transformers  of  a  recent  type. 

Table  of  Transformer  Efficiencies. 

Per  cent. 
Output  Efficiency 

Kilo-watts.  Full  Load. 

I  .    .    . 94-8 

2 95-75 

3 96.2 

4 96.45 

5 96-65 

6 96-73 

7 96.8 

8 96-85 

9 ....'. 96.9 

10 96.95 

IS 97  2 

The  efficiency  of  a  given^  transformer  is  very  low  when  the 
output  is  small,  it  increases  as  the  output  increases,  reaches  a 
maximum,  and  falls  off  again  when  the  output  is  very  great. 
This  falling  off  of  efficiency  when  the  output  is  great  is  due  to 
the  great  increase  of  copper  losses.  Fig.  127  shows  the  effi- 
ciency of  a  transformer  at  various  loads. 

Calculation  of  efficiency. — The  transformer  output  (noninductive 
receiving  circuit)  is  E"I" .  The  internal  loss  is  W^,  +  W^-\-  W^ 
so  that  the  intake  is  E"I"  -^  W^-^  W^-\-  W^  and  the  efficiency  is: 

E"P'  ,     . 

'^  -  E>'i"  ^w,+  w^-j-  w:  ^^'^^ 


TRANSFORMERS. 


139 


All-day  efficiency. — Usually  a  transformer  is  connected  to  the 
mains  continuously,  and  current  is  taken  from  the  secondary  for 
a  few  hours,  only,  each  day.  In  this  case  the  iron  loss  is  inces- 
sant and  the  copper  loss  is  intermittent.  The  total  work  given 
to  the  transformer  during  the  day  may  greatly  exceed  the  total 
work  given  out  by  it,  espe- 
cially if  the  incessant  iron 
losses  are  not  reduced  to  as 
low  a  value  as  possible. 
The  ratio  total  work  given 
out  by  the  transformer  -=- 
total  work  received  by  the 
transformer  during  the  day 
is  called  the  all  day  effi- 
ciency of  the  transformer. 

107.  Practical  and  ulti- 
mate limits  of  output  of 
a  transformer. — When  the 
secondary  current  of  a 
transformer  is  increased  the 
secondary  e.  m.  f.  generally 
drops  off,  and  the  power  output  increases  with  the  current  and 
reaches  a  maximum  as  in  the  case  of  the  alternator.  This 
maximum  power  output  is  the  ultimate  limit  of  output  of  the 
transformer.  Practically  the  output  of  a  transformer  is  limited 
to  a  much  smaller  value  than  this  maximum  output  because  of 
the  necessity  of  cool  running,  because  in  most  cases  it  is  necessary 
that  the  secondary  e.  m.  f  be  nearly  constant,  and  because  the 
efficiency  of  a  transformer  is  low  at  excessive  outputs. 

Small  transformers  have  relatively  large  radiating  surfaces  and 
in  such  transformers  the  requirements  of  close  regulation,  as  a 
rule,  determine  the  allowable  output. 

Large  transformers  have  relatively  small  radiating  surfaces  and 
their  allowable  output  is  limited  by  the  permissible  rise  in  tem- 


/• 

1^ 

.y 

-^ 

^ 

■8 
•7 
•6 

" 

^ 

/ 

/ 

/ 

/ 

/ 

•t 

3 

1 

/ 

•2 

1 

1 

frcn 

i/on 

^c/ 

ful 

Uoc 

J 

/f 


Fig.  127. 


I40      THE   ELEMENTS   OF   ALTERNATING  CURRENTS. 

p'erature.  Very  large  transformers  are  usually  provided  with  air 
passages  through  which  air  is  made  to  circulate  by  a  fan.  Some- 
times transformers  are  submerged  in  oil,  which,  by  convection, 
carries  heat  from  the  transformer  to  the  containing  case,  where  it  is 
radiated. 

Large  transformers  are  much  more  efficient,  under  full  load, 
than  small  ones,  and  give  closer  regulation. 

108.  Rating  of  transformers. — A  transformer  is  rated  accord- 
ing to  the  power  it  can  deliver  steadily  to  a  noninductive  receiving 
circuit  without  undue  heating ;  and  the  ratio  of  transformation, 
together  with  a  specification  of  the  frequency  and  effective  value 
of  the  primary  e.  m.  f  to  which  the  tranformer  is  adapted,  are 
given. 

The  rating  of  a  transformer  is  by  no  means  rigid.  Thus,  if  a 
transformer  is  used  to  give  more  than  its  rated  output  it  will 
become  somewhat  more  heated  by  the  internal  losses  and  its 
regulation  will  not  be  so  close.  If  a  transformer  is  used  for  a 
primary  e.  m.  f.  greater  than  its  rated  primary  e.  m.  f  or  for  a 
frequency  lower  than  its  rated  frequency,  the  range  of  flux  den- 
sity B  in  the  core  will  be  increased,  which  will  increase  the  core 
losses.  Some  manufacturers  rate  their  transformers  generously, 
so  that  they  may  be  greatly  overloaded  or  used  with  greatly  in- 
creased primary  e.  m.  f.  or  decreased  frequency  without  difficulty. 

109.  Outline  of  transformer  design. — A  transformer  is  usually 
designed  to  take  current  from  mains  at  a  prescribed  e.  m.  f  and 
frequency,  and  to  deliver  current-  at  a  prescribed  e.  m.  f.  to  a  re- 
ceiving circuit ;  the  transformer  must  be  so  proportioned  and  of 
such  size  as  to  deliver  the  prescribed  amount  of  current  steadily 
without  undue  heating  and  without  any  great  variation  of  its  sec- 
ondary e.  m.  f  from  zero  to  full  load. 

In  the  designing  of  a  transformer  there  is  but  one  condition 
which  must  be  precisely  met,  namely,  the  ratio  of  primary  to  sec- 
ondary turns  must  be  equal  to  the  ratio  of  the  prescribed  pri- 
mary and  secondary  e.  m.  f.'s.     All  other  points  in  design  are  to 


TRANSFORMERS. 


141 


a  great  extent  matters  of  choice  guided  in  a  general  way  by  ex- 
perience. 

The  accompanying  table  gives  magnetic  flux  densities  which 
are  usually  employed  in  transformer  cores. 

The  allowable  temperature  rise  varies  greatly  with  different 
makers,  the  extent  of  radiating  surface  required  per  watt  of  loss 
per  degree  rise  of  temperature  varies  between  extremely  wide 
hmits,  and  no  simple  rule  can  be  given  covering  this  matter. 

Magnetic  Densities  B  for  Transformer  Cores. 


Frequency. 

Small 
Transformers. 

Medium 
Size  Transformers. 

Large 
Transformers. 

25 

7500 

6750 

6000 

40 

6500 

5750 

5000 

60 

5000 

4750 

4500 

80 

4500 

4250 

4000 

100 

4000 

3750 

3500 

120 

3500 

3250 

3000 

Given  the  required  power  output*  of  a  transformer,  the  value 
and  frequency  of  the  primary  e.  m.  f ,  and  the  value  of  the  sec- 
ondary e.  m.  {.,  the  design  of  the  transformer  is  conveniently  de- 
termined as  follows  : 

Find  from  the  table  the  efficiency  which  can  probably  be  attained 
and  calculate  the  total  transformer  loss  at  full  load.  Of  this  total 
loss  about  half  should  be  iron  loss  and  half  copper  loss.f  The 
total  iron  loss  is  : 

W.^a  VfE"-^  +  b  VfH^B^  (80) 

according  to  equations  (^y6)  and  (77). 

Having  decided  upon  maximum  flux  density  B  (see  accom- 
panying table),  and  upon  thicknessj  of  laminations  /,  equation 

*  Rated  output  is  the  output  which  the  transformer  can  deliver  satisfactorily  to  a 
nonindudive  circuit. 

f  If  the  transformer  is  to  be  connected  to  the  mains  all  day  but  is  to  deliver  current 
only  four  hours  per  day,  for  example,  then  the  iron  loss  during  24  hours  should  be 
about  equal  to  the  copper  loss  during  four  hours  or  under  full  load  the  copper  loss 
should  be  several  times  as  great  as  the  iron  loss. 

%  12  to  16'  thousandths  of  an  inch  is  the  thickness  usually  employed. 


142      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

(80)  gives  the  volume  Fof  iron  to  be  used  in  the  transformer 
core.  The  core  may  be  made  of  the  type  shown  in  Fig.  128  or 
of  the  type  shown  in  Fig.  129.  The  proportions  (relative  dimen- 
sions) indicated  in  Figs.  128  and  129  will  be  found  to  give  satis- 


FiG.  128. 


factory  results  although  the  form  of  the  core  may  be  considerably 
modified  without  greatly  affecting  the  action  of  the  transformer. 
In  fact,  it  is  usually  necessary  to  modify  the  core  slightly  after 
the  coils  have  been  desip;ned. 


■  _i 

-=.^_^ 

i 

-^— 

~— 

■^^ 

=^ 

I    H-^H 


Fig.  129. 

The  maximum  core  flux  N  is  equal  to  the  product  of  the  sec- 
tional area  of  the  magnetic  circuit  (where  it  passes  through  the 


TRANSFORMERS. 


143 


coils)  into  the  maximum  flux  density  B.     Then  equation  (69) 
determines  the  primary  turns,  namely, 


coN 

)r 

- 

Z>  = 

E'lo^ 
'  4A4W 

(81) 

The  number  of 

secondary 

turns  is  then  determined 

by 

equa- 

ion  (62) 

namely. 

Z" 

-  Z'. 

-  E'^ 

(82) 

From  the  provisionally  designed  core  the  mean  length  of  a 
turn  of  primary  and  of  secondary  coils  may  be  determined  which 
together  with  Z'  and  Z"  gives  the  total  lengths  of  wire  in  primary 
and  secondary.  The  size  of  this  wire  is  then  easily  chosen  so 
that  R'l'^  and  R"I"^  may  be  each  equal  to  half  the  full  load 
copper  loss. 

The  size  of  wires  being  thus  determined  the  space  necessary 
for  the  coils  and  insulation  can  be  estimated.  If  the  provision- 
ally designed  core  gives  more  or  less  space  than  is  required  for 
the  coils  its  dimensions  may  be  altered  to  suit. 

Transformer  Connections. 
110.  Simple  connection.     In  parallel.     In  series. — When  used 
to  supply  current  to  lamps  or  motors  from  constant  potential 


main 


main 


I  receiving  \  \  receiving  \ 

circuit  circuit 

Fig.  130. 

mains  the  primary  of  the  transformer  is  connected  to  the  mains 


144      THE   ELEMENTS    OF   ALTERNATING   CURRENTS. 

and  the  secondary  of  the  transformer  is  connected  to  the  terminals 
of  the  receiving  circuit.  When  a  number  of  receiving  circuits 
are  supphed  through  separate  transformers  the  primaries  of  the 
transformers  are  connected  in  parallel,  as  shown  in  Fig.  130. 


-'Wmm^ 


-^ 


■^ 


Fig.  131. 


When  current  is  supplied  through  transformers  to  a  number  of 
arc  lamps  from  a  constant  current  alternator  the  transformer  pri- 
maries are  connected  in  series  and  the  lamps  are  connected  to  the 
respective  secondaries,  as  shown  in  Fig.  131.  This  arrangement 
is  seldom  employed. 


u 


main 


main 


-lOOV- 


o 
o 


-lOOV- 


-eoov- 


o 
o 


Fig.  132. 

111.  Transformers  with  divided  coils. — Alternators  for  isolated 
lighting  plants  give  usually  1000  or  2000  volts  e.  m.  f  and  the 
standard  e.  m.  f.'s  for  incandescent  lamps  are  55  and  no  volts. 


TRANSFORMERS. 


145 


Transformers  are  frequently  made  with  two  primary  coils,  which 
may  be  connected  in  series  for  2000  volts  or  in  parallel  for  looo 
volts,  and  with  two  secondary  coils,  which  may  be  connected  in 
series  to  give  no  volts  or  in  parallel  to  give  55  volts. 

Transformers  for  supplying  current  for  testing  purposes  are 
frequently  made  with  a  number  of  secondary  coils,  which  may  be 
connected  to  p-ive  hig-h  or  low  e.  m.  f 's  as  desired. 


main 


■mam 


mmmmBm 


Fig.  133. 

Transformers  for  supplying  current  to  the  Edison  three-wire 
system. — For  this  purpose  two  similar  transformers  may  be  used 
as  shown  in  Fig.  132,  or  a  single  transformer  with  two  secondary 
coils  may  be  used  as  shown  in  Fig.  133. 


main 


■mam 


■mam 


G^ 


main 


ftRTSlRr^ 


E, 


& 


■mwsw-p 


5> 


— -J 


Fig.  134.  Fig.  135. 

112.  Connecting^  of  primary    and    secondary  coils  in  series. — 

The    ratio  of  transformation    of   a   given   transformer   may   be 
altered  by  connecting  the  secondary  coil  in  series  with  the  pri- 

10 


146      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


mary  as  shown  in  Figs.  134  and  135.     The  arrangement  shown 

E, 
in  Fig.  134  gives  a  ratio  of  transformation  -^  equal  to  ^  =b  i,  and 

E,  I      ,  . 

the  arrangement  shown  in  Fig.  1 3  5  gives  -pr  =  i  zh  -  where  a  is 

the  ordinary  ratio  of  transformation  of  the  transformer.     The  + 
or  —  sign  is  taken  according  as  the  secondary  is  connected  so  as 

to  help  or  so  as  to  oppose 
the  primary.  These  ar- 
rangements are  not  advis- 
able in  commercial  work 
for  the  reason  that  they 
involve  the  connecting  of 
the  low  e.  m.  f.  mains 
directly  to  the  high  e.  m.  f 
mains  which  is  a  source 
of  danger. 

113.  Two -phase  three- 
phase  transformers. — The 
use  of  simple  transformers 
for  two-phase  and  for 
three-phase    systems    has 


^, 


^•J? 


been  explained.     Transformers  are  frequently  used  to  transform 
from  two-phase  to  three-phase  or  vice  versa. 

To  produce  an  e.  m.  f.  of 
any  specified  value  and 
phase. — Let  A  and  B,  Fig. 
136,  be  the  two  e.  m.  f's 
of  a  two-phase  dynamo 
and  let  it  be  required  to 
produce  an  e.  m.  f  E^  of 
given  value  and  phase. 
The  component  of  E^  par- 
allel to  A  is  E^  sin  /?  and  the  component  of  E^  parallel  to  B  is 


Fig.  137. 


TRANSFORMERS. 


147 


E-^  cos  /?.  Fig.  137  shows  two  distinct  transformers  with  simi- 
lar primary  coils  one  connected  to  phase  A  the  other  to  phase 
B.  A  secondary  a  may  be  wound  upon  the  one  transformer 
to  give  the  component  E^  sin  /5  ;  and  a  secondary  b  may  be  wound 
upon  the  other  transformer  to  give  the  other  component  E^  cos  /?. 
Similarly  any  other  e.  m.  f.  such  as  E^  or  E^,  Fig.  136,  may  be 
produced  by  a  pair  of  properly  proportioned  secondary  coils. 

TJie  two-phase  three-phase  transformer  consists  of  two  distinct 
transformers  A  and  B,  Fig.  137,  wound  with  similar  primary 
coils  to  which  the  two-phase  e.  m.  f 's  are  connected.  Each  of 
the  three-phase  e.  m.  f 's  is  (in  general)  generated  in  a  pair  of 
secondary  coils,  one  on  each  transformer.  Such  a  transformer 
transforms    equally   well    from    three-phase    to    two-phase    or 


*»                   w*                     « 

A 

B 

l»,j^ q 

'\     e\  ^y<^ / 

wmm^ 

mmm 

/ 
/ 

/ 

•               1 

a 

mm 

b 

* 

(mmm 

c  . 

\ 

s 
\ 

\ 

V 

! 

2 

Fig. 

138. 

Fig.  139. 

from  two-phase  to  three-phase.  The  three  pairs  of  secondary 
coils  may  be  connected  according  to  the  J  scheme  or  Y  scheme. 
In  the  first  case  the  e.  m.  f.'s  between  the  three-phase  mains  are 
the  e.  m.  f 's  produced  in  the  respective  pairs  of  coils.  In  the 
second  case  the  e.  m.  f 's  between  the  mains  (7n)  are  related  to 
the  e.  m.  f 's  generated  by  the  respective  pairs  of  coils  (c)  as 
shown  in  Fig.  138  and  as  explained  in  Art.  68. 

Consider  any  point  0^,  Fig.  138.  If  a  pair  of  secondary  coils 
is  arranged  on  A  andB,  Fig.  137,  to  give  an  e.  m.  f.  O'p,  another 
pair  to  give  an  e.  m.  f  O'q,  and  a  third  pair  to  give  an  e.  m.  f 


148      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


^. 


^/ 


P' 


\% 


a 


m 


^r 


B 


O'r;  then  these  three  pairs  of  coils  K-connected  would  give  the 
three  phase  e.  m.  f.'s  'tn  ni  m  between  the  mains. 

Scoffs  transformer. — The  simplest  two-phase  three-phase 
transformer  is  that  due  to  Scott.  This  transformer  consists  of 
two  cores  with  similar  primaries  A  and  B,  Fig.  139.  These 
two  primaries  are  connected  to  the  two-phase  mains.  One  core 
has  a  single  secondary  coil  c  and  the  other  has  two  similar  sec- 
ondaries, each  having  ^v^3  times  as  many  turns  as  the  coil  c. 
These  coils  ^5  and  <:are  F-connected  to  the  three-phase  mains  i,  2 

and  3,  as  shown.     The 
point  0' ,  Fig.  138,  lies 
for  Scott's  transformer 
midway    between     the 
points  p  and  r  as  shown 
^      in    Fig.    140.     In    this 
^    figure  a  b  and  c  repre- 
FiG.  140.  sent  the   e.  m.  f.'s  in- 

duced in  the  coils  a  b  and  c,  Fig.  139,  respectively  ;  the  two-phase 
e.  m.  f.'s  A  and  R  being  parallel  to  a  and  to  c  respectively  as 
shown. 

114.  The  monocyclic  system. — The  monocyclic  generator  of 
the  Gen.  Elec.  Co.  is  a  polyphase  dynamo  not  strictly  to  be  called 
two-phase  or  three- 
phase.  It  is  employed 
in  stations  where  a 
small  portion  of  the 
output  is  used  for  mo- 
tors and  a  large  portion 
for  lighting.  The  ar- 
mature winding  of  the 
monocyclic  generator  is 
essentially  a  two -phase 
winding.  The  A  winding  has  four  times  as  many  conductors  as 
the  B  winding,  and  one  end  of  the  B  winding  is  connected  to  the 


TRANSFORMERS. 


149 


J 


r 


B 


middle  point  of  the  A  winding,  as  shown  in  Fig.  141.  The  three 
collecting  rings  are  indicated  by  i,  2  and  3.  Main  3  is  called 
the  teaser. 

Lamps,  or  transformers  feeding  lamps,  are  connected  to  mains 
I  and  2,  and  two  similar  transformers  connected,  as  shown  in  Fig. 
142,  are  used  to  supply  three-phase  currents  to  induction  motors. 

Problems. 

1.  A  given  transformer  is  rated  at  5  kilowatts  and  is  designed 
to  take  current  from  i  lOO-volt  mains  at  a  frequency  of  60  cycles 
per  second.  Under  these  conditions  the  iron  loss  and  the  copper 
loss  will  be  called  normal. 

The  transformer  is  used     Tnc 

at   6    kilowatts    output   at     i32i0SJ::^ 

rated  e.  m.  f.  and  frequency.     — 
Calculate    copper    loss   in 
terms  of  normal. 

The  transformer  is  used 
at  rated  e.  m.  f.  but  at  a 
frequency  of  75  cycles  per 
second.  Calculate  iron  loss 
in  terms  of  normal. 

The  transformer  is  used 
at  rated  frequency  but  with  primary  e.  m.  f.  of  1500  volts.     Cal- 
culate iron  loss  in  terms  of  normal. 

With  primary  e.  m.  f.  of  1 500  volts  what  frequency  would  give 
normal  iron  loss  ? 

With  primary  e.  m.  f.  of  1 500  volts  what  load  would  give 
normal  copper  loss  ? 

2.  The  core  of  a  transformer  has  7200  cubic  centimeters  of 
annealed  refined  sheet  iron  of  thickness  .035  centimeter.  The 
sectional  area  of  the  magnetic  circuit  is  144  square  centimeters 
and  the  length  of  the  magnetic  circuit  is  50  centimeters.  The 
permeability  of  the  iron  is  1250.  The  primary  coil  has  500 
turns  and  is  connected  to  a  i  lo-volt  alternator  having  a  frequency 


a 


Fig.  142. 


ISO   THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

of  125  cycles  per  second.  Calculate  the  following  quantities: 
(a)  Maximum  core  flux.  ((5)  Maximum  flux  density,  (c)  Hys- 
teresis loss  in  watts,  (d)  Eddy  current  loss  in  watts,  (e)  Mag- 
netizing current.  Also  calculate  phase  difference  between  pri- 
mary e.  m.  f.  and  magnetizing  current.  Magnetizing  current  here 
means  the  total  current  in  the  primary  coil,  circuit  of  secondary 
coil  being  of  course  open. 

3.  The  primary  and  secondary  e.  m.  f.  ratings  of  a  transformer 
are  1 100  volts  and  no  volts  respectively.  The  secondary  is  con- 
nected in  series  with  4  ohms  (noninductive)  to  no-volt  mains 
and  the  primary  is  connected  to  a  noninductive  resistance  of  200 
ohms.     Calculate  current  in  each  coil. 

The  primary  (i  loo-volt  coil)  is  in  two  parts  which  are  in  series 
for  1 100  volt  primary  rating.  These  two  parts  are  thrown  in 
parallel,  resistance  being  the  same  as  before.  Calculate  primary 
current  and  secondary  current.  In  this  problem  ignore  the  re- 
sistances of  the  transformer  coils. 

4.  A  transformer  of  which  the  primary  and  secondary  e.  m. 
f.'s  are  rated  at  iioo  and  no  is  connected  to  looo-volt  mains 
with  its  secondary  connected  in  series  with  the  primary.  Calcu- 
late the  e.  m.  f  between  the  terminals  of  the  secondary  coil  {a) 
when  the  secondary  coil  helps  the  primary,  (d)  when  the  secondary 
coil  opposes  the  primary. 

5.  The  primary  coils  of  two  transformers  have  each  560  turns 
of  wire  and  they  are  connected  to  two-phase  mains,  the  e.  m,  f. 
of  each  phase  being  800  volts. 

Calculate  the  turns  of  wire  required  in  each  of  two  secondary 
coils  (one  on  each  transformer)  so  that  these  coils  when  con- 
nected in  series  give  an  e.  m.  f  of  400  volts  30°  ahead  of  one  of 
the  two-phase  e.  m.  f.'s. 


CHAPTER   XII. 


THE   SYNCHRONOUS   MOTOR. 

115.  Two  alternators  in  series. — Consider  two  alternators  A 
and  B  connected  in  series  and  driven  to  give  precisely  the  same 
frequency.  Let  the  lines  A  and  B,  Fig.  143,  represent  the  effec- 
tive e.  m.  f 's  of  machines  A  and  B  respectively  and  let  (p  be  an- 
gular lag  of  the  e.  m.  f.  B  behind  the  e.  m. 
f  A.  The  vector  sum  of  A  and  B,  namely 
E,  is  the  resultant  e.  m.  f  of  the  two  machines 
and  this  e.  m.  f ,  according  to  Prob.  IV.,  Chap. 
v.,  produces  a  current 


/= 


E 


which  lags  d°  behind  E  where  : 

~R 


tan  d  = 


(83) 


(84) 


in  which  R  is  the  total  resistance  of  the  cir- 
cuit, L  its  total  inductance  including  the 
armatures  of  both  machines,  and  en  =  2nf,  f 
being  the  frequency. 

The  power  P'  put  into  the  circuit  by  ma- 
chine A  is  ^'^-  '43- 

P==AI  cos  {AI)  (85) 

when  {AI)  is  the  angle  between  A  and  /.     The  power  P"  put 
into  the  circuit  by  machine  B  is 

P' =  BI  cos  {BI).  (86) 

The  angle  (AI)  in  Fig.  143  is  less  than  90°,  so  that  cos  {AT) 
is  positive ;  therefore  P  is  positive,  that  is,  the  machine  A  is  act- 

(151) 


152       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

ing  as  a  dynamo.  The  angle  (BI)  in  the  figure  is  greater  than 
90°,  so  that  cos  {BI)  is  negative ;  therefore  P"  is  negative,  that 
is,  the  machine  B  is  acting  as  a  motor. 

The  alternator  B  used  in  this  way  is  called  a  synchronous  motor, 
the  alternator  A  being  driven  by  an  engine  or  water  wheel. 

116.  Variation  ai  P  and  P"  with  the  Phase  Ang-le  tp. — Draw  a 
line  OC,  Fig.  144,  representing  B  to  scale.     Describe  about  C  2. 


Fig.  144. 


circle  of  which  the  radius  represents  A.  Then  a  line  OP^  from 
0  to  any  point  in  the  circle  represents  a  possible  value  of  the  re- 
sultant e.  m.  f.  E,  (f  being  the  corresponding  phase  difference  be- 
tween A  and  B.  Draw  the  line  ef  through  0,  making  with  OC 
the  angle  6.  Then  the  angle  POf  is  equal  to  the  angle  (BI). 
Therefore 


OQ  =  OP  cos  (BI)  =  E  cos  {BI)  =  s/R'  -f  o)'L^  I  cos  {BI] 


THE   SYNCHRONOUS   MOTOR. 


153 


or  /  cos  {BI)  = 


00 


I  cos  {BI)  in  equation  {S,6)  we  have 

B.'OQ 


Substituting     this     value     of 


P"  = 


That  is,  the  output  of  the  machine  B  is  proportional  to  the 
projection  OQ  of  the  line  OP  on  the  line  ef,  B  and  s/ E?  +  a?I} 
being  constant. 

When  Q  is  towards /from  O  cos  {BI)  is  positive  so  that  /^"  is 
positive  and  machine  B  acts  as  a  dynamo.  When  Q  is  towards 
e  from  0  cos  {BI)  is  negative  so  that  P"  is  negative  and  machine 
B  acts  as  a  motor. 


ou 


Fig.  145  is  a  construction,  for  the  same  value  of  <p,  m  which 
OQ!  represents  the  power  P'  put  into  the  circuit  by  machine  A. 
From  this  diae^ram  we  have 


154      THE    ELEMENTS   OF   ALTERNATING   CURRENTS. 


P'=-jt£^=.     .  (88) 

The  projection  of  A,  Fig.  144,  on  ^  is  ^  cos  ((p—d);  the  pro- 
jection of  B  on  ef  is  B  cos  d ;  and  OQ  is  the  sum  of  these  pro- 
jections so  that 

OQ  =  A  cos  {(p  —  d)-\.B  cos  d. 

Substituting  this  value  of  OQ  in  equation  (87)  we  have 


P"  = 


cos  {<p-d)+    ,_   _      _,  cos  d.      (89) 


Similarly  from  Fig.  145  we  have 

A'R  A^ 

P'  =  ...  cos  {<p  +  e)+  __    ,^,  cos  e.    (90) 


>/ie^  +  (O^D 


VK'  +  oj^D 


~to       to       5o      7Jo     755      780      575     2?o     373     jtfo     330     j(,o 

rKase  Xiifference    ^ 

Fig.  146. 

Further  the  algebraic  sum  of  the  outputs  of  machines  A  and 
B,  namely,  P'  -f  P"  is  equal  to  RP  so  that 

P'  +  P"  =  RP.  (91) 

These  three  equations,  (89),  (90)  and  (91)  are  the  fundamental 
equations  of  the  synchronous  motor. 

The  ordinates  of  the  curves  P,  P'  and  RP,  Fig.  146,  show 


THE   SYNCHRONOUS   MOTOR. 


155 


the  values  of  P  of  P"  and  of  RP  for  values  of  ^  from  zero  to 
360°.  Positive  ordinates  represent  positive  power  (dynamo  ac- 
tion), negative  ordinates  represent  negative  power  (motor  action). 
Each  ordinate  of  the  curve  RP  is  the  algebraic  sum  of  the  ordi- 
nates of  the  curves  P',/"^  Fig.  147  shows  portions  of  the  curves 


P' ,  P"  and  RP  to  a  larger  scale.  The  ordinates  of  the  curve 
rj  represent  the  efficiency  1-^)  of  transmission  for  various  val- 
ues of  <p  when  machine  j5  is  a  motor. 

Figs,  143  to  148  are  based  on  the  values  A=  iioo  volts, 
^=  1000  volts,  R=i  ohm  and  coL  =  0.58  ohm,  also  the  par- 
ticular statements  given  below  are  based  on  these  values. 

By  comparing  the  ordinates  of  the  curves  P'  and  P'',  Fig.  146 
or  Fig.  147,  it  is  seen  that  when  machine  ^  is  a  motor  (negative 
values  of  P)  its  intake  is  very  much  less  than  the  output  of  P. 
Therefore  the  efficiency  of  transmission  is  quite  small  when  the 


156       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

machine  A,  having  the  larger  e.  m.  f.,  is  the  motor.*  The  maxi- 
mum intake  of  machine  ^  as  a  motor  is  45,1  kilowatts.  The 
maximum  intake  of  machine  ^  as  a  motor  is  202.6  kilowatts. 

When  machine  B  is  running  as  a  motor,  unloaded,  its  intake 
P'  is  approximately  zero  ;  the  point  P,   Fig.  148,  is  at  s,^  and 


^^r^oLA 


Fig.  I 


the  value  of  ^  is  171°. 9.  When  the  intake  of  B  is  zero  the  re- 
sultant e.  m.  f  is  Os,  Fig.  148  ;  the  current,  which  is  in  quadrature 
with  j5,  is  about  140  amperes  ;  RI^  is  about  20  kilowatts  ;"and  the 

*  When  the  reactance  uL  of  the  circuit  is  large  compared  with  R,  then  the  ma- 
chine having  the  larger  e.  m.  f.  may  be  used  as  the  motor  without  greatly  reducing 
the  efficiency  of  transmission  ;  also  when  the  reactance  of  the  circuit  is  great  compared 
with  R  the  e.  m.  i.  of  the  motor  may  greatly  exceed  the  e.  m.  f.  of  the  generator  as 
will  be  seen  later. 

f  Running  of  B  is  unstable  when  P  is  at  S^. 


THE   SYNCHRONOUS   MOTOR,  157 

output  of  A  is  of  course  equal  to  RI^.  As  the  motor  B  is  loaded 
the  point  P,  Fig.  148,  moves  from  s  towards  M;  and  the  result- 
ant e.  m.  L,  also  the  current  and  RI",  grow  less  until  ^  =  180°. 
The  minimum  value  of  /  is  about  go  amperes,  and  the  minimum 
value  of  RI^  is  about  8.  i  kilowatts.  When  the  point  P,  Fig.  148, 
reaches  the  line  ef  the  current  is  opposite  to  the  e.  m.  f.  B  in 
phase  and  the  efficiency  of  transmission  is  the  greatest  possible 
for  the  given  values  of  A,  B,  coL  and  R.  This  maximum  effi- 
ciency is  about  92^  and  the  value  of  <p  is  182°. 9.  As  the  motor 
B  is  still  further  loaded  the  point  P  moves  on  towards  M  and 
when  P  reaches  M  the  intake  of  B  is  at  its  maximum  value  of 
202.6  kilowatts.  Further  loading  of  B  causes  it  to  fall  out  of 
synchronism  and  stop.  In  practical  working  the  e.  m.  f  of  the 
dynamo  {A)  is  usually  greater  than  that  of  the  motor  {B),  and 
the  load  is  limited  to  a  smaller  value  than  the  maximum  for  the 
sake  of  high  efficiency  and  of  stability. 

Remark :  When  the  polyphase  alternator  is  used  as  a  synchron- 
ous motor  each  armature  winding  of  the  machine  receives  current 
from  one  phase  of  a  polyphase  system,  and  the  total  power  in- 
take is  two  times  or  three  times  as  great  as  the  intake  of  each 
winding,  according  as  the  machine  is  a  two-phase  or  a  three- 
phase  machine.  The  present  chapter  deals  explicitly  with  the 
single-phase  synchronous  motor,  but  the  entire  discussion  applies 
equally  well  to  the  polyphase  machine.  For  example  :  OC,  Fig, 
144,  may  represent  the  e.  m.  f  of  one  winding  of  a  polyphase  syn- 
chronous motor  and  A  the  e.  m.  f.  of  one  phase  of  the  supply 
dynamo,  then  P"  equation  (89)  is  one -half  or  one-third  of  the 
total  power  intake  of  machine  B  and  i?/^  equation  (91)  is  the 
watts  lost  in  one  circuit  only  of  the  polyphase  system. 

117,  The  starting  of  the  synchronous  motor. — To  get  the  ma- 
chine B  into  operation  as  a  synchronous  motor  it  must  *  be  started 
by  some  independent  mover,  by  an  induction  motor,  for  example, 

*  A  polyphase  synchronous  motor  is  an  induction  motor  when  running  below  syn- 
chronism and  is  self-starting. 


158       THE   ELEMENTS   OF   ALTERNATING   CURRENTS, 

and  its  speed  carefully  adjusted  until  it  is  in  synchronism  with  the 
machine  A,  which  has  been  started  up  beforehand.  So  long  as 
the  frequency  of  B  is  less  than  the  frequency  of  A  the  angle  <f, 
Figs.  143  and  144,  increases  continuously,  the  point  P,  Fig.  144, 
passes  once  around  the  circle  while  A  gains  one  cycle  on  B,  and, 
if  the  circuit  contains  no  auxiliary  resistance,  the  outputs  of  A  and 
B,  namely  P'  and  P',  pass  repeatedly  through  the  series  of  values 
shown  in  Fig.  146.  During  the  time,  however,  that  the  machine 
B  is  being  adjusted  to  synchronism  with  A  an  auxiliary  resist- 
ance, usually  a  lamp,  is  connected  in  circuit,  as  shown  in  Fig. 


Fig.  149. 


149,  to  limit  the  excessive  values  that  would  otherwise  be  reached 
by  P'  and  P".  This  lamp  pulsates  in  brightness  as  machine  B 
is  being  speeded  up,  and  the  pulsations  become  slower  and  slower 
as  the  frequency  of  B  approaches  the  frequency  of  A.  When  the 
lamp  is  brightest,  (p=o°  or  360°,  and  when  the  lamp  is  dimmest, 
^=180°,  as  is  shown  by  the  R/^  curve.  Fig.  146.  When  the 
machines  are  very  nearly  in  synchronism  the  pulsations  of  the 
lamp  are  very  slow,  and  the  switch  s  is  then  closed  as  the  lamp 
passes  its  minimum  of  brightness.  The  machine  B  is  then  in 
operation  as  a  synchronous  motor  and  the  independent  mover 
used  to  start  B  may  be  disconnected. 

In  practice  the  lamp  /  is  connected  in  series  with  the  second- 
aries of  two  transformers,  the  primaries  of  which  are  connected 
to  A  and  to  B  respectively.  In  this  arrangement  the  lamp  may 
be  either  at  its  maximum  or  at  its  minimum  of  brightness  when 


THE   SYNCHRONOUS   MOTOR.  1 59 

the  proper  conditions  are  reached  for  the  closing  of  the  switch  s, 
according  to  the  connections  of  the  transformers. 

118.  Stability  of  running^  of  synchronous  motor. — Suppose  the 
machine  A  to  be  driven  by  means  of  a  governed  engine  at  con- 
stant speed,  irrespective  of  its  output ;  and  suppose  the  motor  B 
to  be  running  steadily.  If  the  load  on  B  is  suddenly  increased 
this  machine  will  run  momentarily  slower  than  A  and  fall  behind 
A  in  phase.  If  this  falling  behind  in  phase  increases  the  power 
which  B  takes  from  A,  then  B  will  fall  only  so  far  behind  as  to 
enable  it  to  take  in  power  sufficient  to  carry  its  increased  load. 
If,  on  the  other  hand,  this  falling  behind  in  phase  decreases  the 
power  which  B  takes  from  A,  then  B  will  fall  further  and  further 
behind  A,  fall  out  of  synchronism  and  stop.  In  the  first  case 
the  running  of  B  is  stable,  in  the  second  case  it  is  unstable. 

As  the  point  P,  Fig.  144,  moves  along  the  circle  in  the  direc- 
tion of  the  arrow  co  the  e.  m.  f.  A  is  getting  farther  ahead  of 
B  in  phase  or  B  is  falling  behind  A.  If  the  motor  intake  of  B  in- 
creases as  it  falls  behind  A,  then  the  running  of  B  is  stable,  and 
vice  versa,  as  pointed  out  above.  Now  the  projection  of  OP, 
namely  OQ,  positive  towards  e,  represents  the  intake  oi  B ;  and 
this  intake  increases  from  s  to  M,  Fig.  148,  as  B  falls  behind  A, 
and  decreases  from  M  to  s'  as  B  falls  behind  A  ;  therefore  s  to  M 
is  the  region  of  stable  motor  running  of  B  and  M  to  s'  is  the 
region  of  unstable  motor  running  of  B. 

If  B  is  running  with  given  load  as  a  motor  the  point  P  will 
take  up  a  position  between  s  and  Ivl  such  that  the  intake  of  B  is 
sufficient  to  carry  its  load.  If  B  is  further  loaded  P  moves 
further  towards  M;  if  B  is  unloaded  P  moves  towards  s.  If  B 
is  loaded  until  P  reaches  M  then  further  loading  decreases  the 
intake  of  B  and  the  machine  B  therefore  falls  out  of  synchronism 
and  stops.  The  action  while  stopping  is  as  follows :  Every 
time  B  loses  one  cycle  as  compared  with  A  the  point  P  moves 
once  around  the  circle,  Fig.  148.  While  P  moves  from  s' 
through  D  to  s  machine  B  acts  as  a  dynamo  which  action  to- 


l6o      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

gather  with  its  belt  load  as  a  motor  slows  it  up  rapidly.  Then  as 
P  moves  from  s  through  M  \.o  s'  machine  B  takes  in  power  from 
A  but  by  no  means  enough  to  enable  it  to  regain  its  speed,  and 
so  on. 

Remark :  For  given  intake  of  ^  (given  value  of  OQ,  Fig.  144) 
there  are  two  values  of  the  resultant  e.  m.  f.  OP  and,  therefore, 
two  values  of  current  /.  The  lesser  value  of  /  corresponds  to 
stable  running  and  the  greater  value  to  unstable  running. 

Rimning  of  two  alternators  in  series  as  dynamos. — Suppose  alternator^,  Fig.  1 50, 
to  be  driven  at  constant  speed  by  a  governed  engine,  and  alternator  B  to  be  driven  by 
an  ungovemed  engine.     Alternator  B  being  once  adjusted  to  synchronism  vi'ith  A 

keeps  in  synchronism  and  gives  a  constant  output. 
The  variations  of  total  load  are  met  by  the  alter- 
nator A  and  its  governed  engine.  The  point  P, 
Fig.  148,  stands  somewhere  between  j-  and  D,  for 
in  this  region  the  output  of  B  decreases  as  it  falls 
behind  and  increases  as  it  gets  ahead  of  A  so  that 
the  ungovemed  engine  and  alternator  B  axe  in 
stable  state  of  running.  Alternators  are  never 
used  thus  in  practice  for  the  reason  that  the  resul- 
tant e.  m.  f.  OB,  Fig.  148,  varies  greatly  with 
the  total  load. 

Ru7i7iing  of  two  alternators  in  parallel  as 
dynamos. — Two  similar  alternators  connected  in 
parallel,  as  shown  in  Fig.  151,  run  satisfactorily 
when  they  are  once  adjusted  to  synchronism  and  this  arrangement  is  frequently  used 
in  practice.  Machine  A  is  started  and  connected  through  a  resistance  /to  B ;  ma- 
chine B  is  then  started  and  when  the  pulsations  of  the  lamp  become  very  slow  the 


>;  Lamps 


Fig.  150. 


Tnam 


Fig.   151. 

switch  s  is  closed  at  minimum  brightness.  The  two  dynamos  together  are  then  con- 
nected to  the  mains.  When  two  machines  are  run  in  parallel  in  this  way  the  machine 
which  pushes  ahead  in  phase  takes  a  larger  portion  of  the  load  and  the  running  is 
therefore  stable.     One  of  the  alternators  should  be  driven  by  an  ungovemed  engine. 


THE   SYNCHRONOUS   MOTOR. 


l6l 


119.  Greatest  intake  of  machine  B ;  A,  B,  coL  and  R  being 
given.  P"  has  its  maximum  negative  value  when  cos  (^  —  ^)  = 
—  I  and  equation  (89)  becomes 

^2  cos  d  -AB 

Fig.  152  shows  the  state  of  affairs  when  intake  of  B  is  at  its 
.  greatest. 


Fig.  152. 

120.  Greatest  value  of  the  e.  m.  f  B  for  which  machine  B  can 
act    as    a    motor ;    A,   coL    and  R   being   given. — So    long  as 

AB  ^2 

■—=====.  IS  greater  than  r=^'    cos  d    then  P"    can 

have  negative  values  according  to  equation  (89).     Therefore  the 

,.    ..  .         ,  AB  B^ 

limitmg    case    is    where 


VR^  +  io^D      s/R"-  +  oi^L" 

A 


cos  u  or 


B  = 


cos  d 


(93) 


II 


1 62       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

This  limiting  case  is  shown  in  Fig.  153. 

121.  To  find  value  of  B  for  which  the  machine  B  may  take  in 
the  greatest  possible  power  from  A  ;  A,  cuL  and  R  being  given. — 
Equation  (92)  expresses  the  greatest  intake  of  B  for  given  values 
of  A,  B,  oiL  and  R.  It  is  required  to  find  the  value  of  B  which 
will  make  this  greatest  intake  a  maximum.  This  value  of  B 
must  render  B'^  cos  d  —  AB  [the  numerator  of  right  hand  mem- 
ber of  equation  (92)]  a  maximum.  Differentiating  this  expres- 
sion with  respect  to  B  and  placing  the  differential  coefficient 
equal  to  zero  we  have 

A  —  2B  cos  ^  =  o 

or  B= (94) 

2  cos  u 

Remark :  A  comparison  of  equations  (93)  and  (94)  shows 
that  the  value  of  B  for  greatest  possible  intake  of  machine  B  is 


Fig.  153. 

half  the  greatest  value  of  B  for  which  machine  B  can  act  as  a 
motor  at  all.  This  is  also  the  case  with  a  direct-current  motor. 
The  greatest  e.  m.  f  such  a  motor  can  have  is  the  e.  m.  f  of  the 
dynamo  which  drives  it,  and  the  value  of  its  e.  m.  f  which  permits 
the  greatest  possible  intake  is  one-half  the  e.  m.  f  of  the  dynamo 
which  drives  it. 


THE   SYNCHRONOUS   MOTOR. 


163 


122.  Value  of  B  to  give  maximum  intake  of  machine  B  with  given 

current ;  A,  o)L  and  R  being  gw en. — Let  /,  Fig.  1 54,  be  the  given 
current  and  E{=  I V K-  +  co^D)  the  resultant  e.  m.  f.  In  order 
that  the  intake  of  B  may  be  a  maximum  BI  cos  (^/)  or  B  cos 
{BI^  must  be  a  maximum.  Now  B  cos  {BI^  is  the  projection 
of  B  on  the  current  line  01.  Describe  a  circle,  center  at  P,  ra- 
dius A  ;  then  Ox  is  the  greatest  possible  value  oi  B  cos  {BI')  for 
the  given  current  and  OC'xs  the  required  value  of  B.  From  the 
triangle  OPC  we  have 


B''  =  A^^E'-2AE  cos 


(95) 


123.  Excitation  characteristics. — ^With  given  load  on  a  syn- 
chronous motor  (given  value  of  P"^  its  e.  m.  f  B  may  be  changed 
by  varying  its  field  excita- 
tion, and  for  each  value  of  B 
there  is  a  definite  value  of  the 
current  /.  Thus  the  abscissas 
of  the  curves,  Fig.  155,  repre- 
sent values  of  /  and  ordinates 
represent  values  of  B,  for 
loads  of  zero,  100  kilowatts 
and  200  kilowatts  respec- 
tively. These  curves  are 
called  the  excitation  charac- 
teristics of  the  motor.  Fig. 
1 5  5  is  based  on  the  values 
A=  1 100  volts,  R=  I  ohm 


Fig.   154. 


and  (t)L  =0.58  ohm.  For  the  greatest  possible  intake,  302.7  ki- 
lowatts, the  characteristic  reduces  to  the  point  enclosed  in  the 
small  circle.  It  was  pointed  out  in  Art.  1 1 8  that  with  given  load 
there  are  two  values  of  /  for  each  value  of  B  and  that  the  larger 
value  of  /  corresponds  to  unstable  and  the  smaller  value  to 
istable  running.  The  dotted  portions  of  the  curves,  Fig.  155, 
correspond  to  the  larger  values  of  /.     These  dotted  portions 


1 64       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


cannot,  of  course,  be  determined  by  experiment  on  account  oif 
the  instability  of  running. 

The  equation  to  the  excitation  characteristics  may  be  derived  as  follows  :  Let  I, 
Fig.  156,  represent  the  current  and  E  the  resultant  e.  m.  f.  ;  the  components  of  E 
are-  RI  and  wZ/.     The  e.  m.  f.  E  is  the  vector  sum  of  A  and  B  as  shown  and  the 

component  of  B  parallel  to  7  is  -r  •     From  the  right-angled  triangles  of  the  figure 
we  have  B"^  —  x"^  ■\-  {  '-^  \  (a) 


tZoo 


lloo 
JOoo 

800 

loQ 

600 
5oo 
4-00 
i3oo 
£00 
loo 


^2: 


(ri^-P^  -{-{x—uL/y 


(b) 


2?a 

1 

-^ 

V 

^ 

^" 

ifoM 

1^ 

">.. 

\, 

^v 

\ 

< 

W^' 

ie 

'^> 

\ 

V 

\] 

X 

^■v 

N, 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 
\ 

\ 

CD 

\ 

^N 

k 

\ 
\ 
\ 

\ 

1 

I 

\ 

li. 
0 

\ 

\ 

k 

J2.7  K.\ 

^. 

\ 
I 

5 

\ 

\ 

\ 

N 

^^ 

\ 

/ 

/ 

^ 

s\ 

1 

/ 
/ 

/ 
/ 

\ 

\ 

/ 

/ 

/ 
/ 

v\ 

/ 

f 

Amp 

EBES 

V 

/OO        Zoo      ioa      4-O0      tfoo       600        7oo       800      9oo       lOoo      llQO     /Zd<y 


Fig.   155. 

By  eliminating  x  from  these  equations  we  have  the  required  relation  between  B  and 
/;  P",  A,  R  and  wZ  being  given.  The  curves,  Fig.  155,  were  calculated  graph- 
ically by  means  of  the  diagram,  Fig.  144. 

124.  Value  of  B  to  bring  I  into  phase  with  A ;  P^',  A,  R  and 
o)L  being  given. — When  /  is  in  phase  with  A,  the  output  of  A  is 


THE    SYNCHRONOUS    MOTOR. 


165 


the  greatest  possible  for  the  given  current.  Let  01,  Fig.  157, 
be  the  current  and  E  the  resultant  e.  m.  f.;  A  being  parallel  to 
/.     The    component    of 


B  parallel  to 


-I    IS    —~. 


From    this   diagram   we 
have 


and 


^  +  RI^  A. 


Eliminating  /  from  these  equations  we  have  an  equation  which 
expresses  the  required  value  of  B  in  terms  of  P",  A,  R  and  wL. 

125.  Value    of    B    to 

bring  I  into  phase  with 

B;     P",  A,  R  a?id  wL 

being  given. — When  /  is 

in  phase  with  B  the  in- 

^^^-  '57-  take  of  B  is  the  greatest 

possible    for   the    given  values  of  B  and  /. 

From  Fig.  158  we  have 

{B  +  Riy  +  oi-'DP  =  A 
Further  P"  =  BI. 

Eliminating  /from  these 
equations  we  have  an  equa- 
tion which  expresses  the 
required  value  of  B  in  terms 
of  P",  A,  R  and  mL. 

Remark :  The  discussion  in  this  chapter  applies  to  the  rotary 
converter  which  is  a  synchronous  motor  in  every  essential 
particular. 


(a) 

(b) 


CHAPTER   XIII. 
THE   ROTARY  CONVERTER. 

126.  The  rotary  converter. — An  ordinary  direct  current  dynamo 
may  be  made  into  an  alternator  by  providing  it  with  collecting 
rings,  as  described  below,  in  addition  to  its  commutator.  Such 
a  machine  is  called  a  rotary  converter. 

The  single-phase  converter  is  provided  with  two  collecting  rings 
which,  in  case  of  a  two-pole  machine,  are  connected  to  diametric- 
ally opposite  armature  conductors. 

The  two-phase  converter  is  provided  with  four  collecting  rings 
which,  in  case  of  a  two-pole  machine,  are  connected  to  armature 
conductors  90°  apart. 

TJie  three-phase  converter  is  provided  with  three  collecting  rings 
which,  in  case  of  a  two-pole  machine,  are  connected  to  armature 
conductors  120°  apart. 

Remark :  It  is  often  convenient  to  refer  to  a  rotary  converter 
as  a  two-ring,  three-ring,  four-ring,  or  ;z-ring  converter,  as  the 
case  may  be. 

Remark :  In  case  of  a  multipolar  mactiine  the  n  collecting 
rings  are  connected  to  the  armature  as  follows  :  Ring  No.  i  is 
connected  to  all  armature  conductors  which,  for  any  given  posi- 
tion of  the  armature,  lie  midway  under  the  north  poles  of  the 
field  magnet.  Let  /  be  the  distance  between  adjacent  conductors 
of  this  first  set,  that  is,  the  distance  from  north  pole  to  the  next 
north  pole.  Then  ring  No.  2  is  connected  to  the  armature  con- 
ductors which  are  -th  of  /  ahead  of  the  first  set ;  ring  No.  x  is 

2 
connected  to  the  armature  conductors  which  are  -ths  of /ahead 

n 

of  the  first  set ;  ring  No.  4  is  connected  to  the  conductors  which 

(166) 


THE   ROTARY  CONVERTER. 


167 


3 


are  -  ths  of  /  ahead  of  the  first  set  and  so  on. 
n 


This  statement 

appHes  to  muhicircuit  winding.  In  case  of  the  two-circuit  wind- 
ing each  collecting  ring  is  connected  to  one  armature  conductor 
only.  Fig.  159  shows  a  four-pole  dynamo  with  two  collecting 
rings  each  connected  to  two  armature  conductors.  The  machine 
when  provided  with  these  collecting  rings  is  a  four-pole  single- 
phase  rotary  converter. 


Fig.  159. 

Use  of  the  rotary  converter. — The  rotary  converter  may  be  used 
as  an  ordinary  direct-current  dynamo  or  motor ;  as  an  alternator 
or  synchronous  motor  ;  it  may  be  driven  as  a  direct-current  motor 
the  load  being  provided  by  taking  alternating  current  from  its 
collecting  rings  ;  or  it  may  be  driven  as  a  synchronous  alternating 
current  motor,  the  load  being  provided  by  taking  direct  current 
off  the  commutator.  This  last  is  the  principal  use  of  the  machine. 
In  every  case  in  which  power,  transmitted  to  a  distance  by  alter- 
nating current,  is  to  be  used  in  the  form  of  direct  current  the 
rotary  converter  is  used  for  bringing  about  the  conversion  from 
alternating  current  to  direct  current.     Thus,  in  many  extended 


l68       THE   ELEMENTS   OF   ALTERNATING  CURRENTS. 

electric  railway  plants  it  is  found  to  be  expedient  to  transmit  the 
power  high  pressure  polyphase  from  a  central  station  to  rotary 
converters  stationed  along  the  line  of  the  railway ;  these  rotary 
converters  in  their  turn  supply  direct  current  at  medium  pressure 
to  the  trolley  wires. 

127.  The  starting  of  the  rotary  converter  and  its  operation  when 
used  to  convert  alternating  current  into  direct  current. — When 
used  in  this  way  the  rotary  converter  is  a  synchronous  motor  and 
it  differs  but  little  in  its  operation  from  the  synchronous  motor 
with  a  belt  load. 


Starting. — The  machine  may  be  started  as  a  direct-current 
motor  using  storage  batteries  or  other  local  source  of  direct  cur- 
rent ;  or  it  may  be  started  in  precisely  the  same  manner  as  a 
synchronous  motor  with  a  belt  load  as  described  in  Art.  117. 
The  field  magnet  of  a  rotary  converter  is  always  excited  by  direct 
current  taken  from  the  machine  itself. 

Operation. — Let  B,  Fig.  160,  be  the  effective  alternating  e.  m. 
f.  of  a  rotary  converter  and  A  the  e.  m.  f  of  the  alternating  gen- 
erator.    Fig.  160  is  identical  to  Fig.  148,  Chapter  XII.     When 


THE   ROTARY   CONVERTER.  1 69 

no  direct  current  is  taken  from  the  converter  its  load  is  zero  and 
the  point  P,  Fig.  1 60,  is  at  s.  When  direct  current  is  taken  from 
the  converter  the  point  P  moves  towards  M.  The  alternating 
current  taken  by  the  converter,  being  proportional  to  the  result- 
ant e.  m.  f.  OP,  at  first  decreases  and  then  increases*  as  the  direct 
current  load  increases,  and  so  on,  exactly  as  in  case  of  a  syn- 
chronous motor  with  a  belt  load.     See  Art.  116. 

It  was  pointed  out  in  Chapter  XII  that  a  synchronous  motor 
may  operate  at  comparatively  high  efficiency  for  a  wide  range  of 
values  of  B  (value  of  A  given)  if  the  alternating-current  circuit 
has  large  reactance ;  in  fact,  B  may  even  be  larger  than  A,  as 
pointed  out  in  Art.  120.  If  a  considerable  portion  of  the  react- 
ance is  external  to  the  armature  of  the  converter  then  the  e.  m. 
f.  between  the  collecting  rings  of  the  converter  changes  with  B 
and  so  also  does  the  direct  e.  m.  f  of  the  machine.  Therefore, 
the  direct  e.  m.  f  of  a  rotary  converter  may  be  varied  at  willf  by 
changing  the  field  excitation  of  the  machine,  although  the  e.  m. 
f  A  of  the  alternating  generator  may  be  constant. 

128.  Pumping  or  hunting  action  of  synchronous  motors  and 
rotary  converters. — When  the  load  on  a  synchronous  motor  is 
increased  the  motor  slows  up  momentarily  and  falls  behind  the 
generator  in  phase.  When  the  motor  has  fallen  behind  sufficiently 
to  take  in  power  enough  to  enable  it  to  carry  its  load  it  is  still 
running  shghtly  below  synchronism  ;  it  therefore  falls  still  further 
behind  and  takes  an  excess  of  power  from  the  generator  which 
quickly  speeds  it  above  synchronism.  It  then  gains  on  the  gen- 
erator in  phase  until  it  takes  in  less  power  than  is  required  for 
its  load,  when  it  again  slows  up  and  so  on.  This  oscillation  of 
speed  above  and  below  synchronism,  called  Jiunting^  is  similar  to 
the  hunting  of  a  governed  steam  engine.  It  is  frequently  a  source 
of  great  annoyance,  especially  where  several  synchronous  motors 
or  rotary  converters  are  run  in  parallel  from  the  same  mains. 

*  When  B  is  less  than  A. 

f  The  possible  range  of  variation  depends  upon  the  reactance  in  the  circuit  ex- 
ternal to  the  rotary  converter. 


I/O      THE   ELEMENTS   OF  ALTERNATING  CURRENTS. 


129.  Armature  current  of  a  rotary  converter. — Consider  a  given 
armature  conductor  of  a  rotary  converter,  A  part  of  the  cur- 
rent in  this  conductor  is  due  to  the  alternating  currents  which 
flow  into  the  armature  at  the  collecting  rings  and  a  part  is  due  to 
the  direct  current  flowing  out  of  the  armature  at  the  direct-cur- 
rent brushes.  The  actual  current  in  the  conductor  is  the  alge- 
braic sum  of  these  two  parts,  and  since  these  parts  are  generally 
opposite  in  sign,  therefore  the  actual  current  in  the  conductor  is 
rather  small  and  so  also  is  its  magnetic  effect  and  its  heating 
effect. 


Fig.  i6i. 

130.  Magnetic  reaction  of  the  armature  of  the  rotary  converter. 

Distortion  of  field. — The  distortion  of  the  magnetic  field  of  a 

dynamo  by  the  armature 
currents  accompanies,  and 
is  in  fact  the  cause  of,  the 
torque  with  which  the  field 
acts  upon  the  armature. 
When  the  torque  is  in  the 
direction  of  the  rotation  of 
the  armature  (motor  action) 
the  field  is  concentrated 
under  the  leading  horns  of 
the  pole  pieces  as  shown  in  Fig.  i6i.  When  the  torque  is 
opposite  to  speed  the  fi.eld  is  concentrated  under  the  trailing 
horns  of  the  pole  pieces  as  shown  in  Fig.  162. 


Fig.  162. 


THE   ROTARY   CONVERTER.  I/I 

When  a  rotary  converter  is  running  steadily  the  speed  of  its 
armature  is  constant,  and  the  only  torque  acting  on  the  armature 
is  the  slight  torque  needed  to  overcome  friction,  therefore  the 
field  is  scarcely  at  all  distorted. 

When  a  rotarj^  converter  hunts,  its  speed  oscillates  above  and 
below  synchronism  so  that  a  torque  acts  upon  the  armature,  first 
in  one  direction  and  then  in  the  other,  and  the  field  is  concen- 
trated, first  under  the  trailing  horns  and  then  under  the  leading 
horns  of  the  pole  pieces. 

Demagnetizing  action. — The  demagnetizing  action  of  the  ar- 
mature currents  of  a  rotary  converter  may  be  considered  as  made 
up  of  the  demagnetizing  action  of  the  direct  current  alone  and  of 
the  alternating  currents  alone.  The  first  is  the  same  as  in  the 
direct-current  dynamo  and  the  second  is  considered  in  Art.  74. 

131.  Power  rating  of  rotary  converters. — The  magnetic  action 
(demagnetizing  action  and  distorting  action)  of  the  armature  cur- 
rents of  a  rotary  converter  is  never  troublesome,  so  that  the  al- 
lowable output  is  limited  by  the  permissible  heating  of  the  arma- 
ture. The  armature  heating  is  rather  small,  as  pointed  out  in 
Art.  1 29,  so  that  a  given  machine  has  a  higher  power  rating  as 
a  rotary  converter  than  as  a  direct-current  dynamo,  except  in  the 
case  of  the  single-phase  converter.  The  accompanying  table 
gives  the  power  ratings  (based  upon  equal  average  armature 
heating)  of  a  given  machine  when  used  [a)  as  a  direct-current 
dynamo,  {b)  as  a  single-phase  converter,  {c)  as  a  three-phase  con- 
verter, ((^)  as  a  two-phase  (four-ring)  converter,  and  i/)  as  a  six- 
phase  converter. 

Power  Ratings  of  Rotary  Converters.* 
a.  b.  c.  d.  e. 

Continuous-  Single-  Three-  Four-  Six- 
current                    phase  phase  ring  phase 
dynamo.  converter.  converter.  converter.  converter. 
1. 00                        .85  1.32  1.62  1.92 

*  These  ratings  are  calculated  as  explained  in  Art.  134,  and  in  their  calculation  the 
losses  in  the  machine  and  the  wattless  component  of  the  alternating  currents  have 
been  ignored.     These  ratings  are  therefore  five  or  six  per  cent,  too  large. 


172      THE   ELEMENTS    OF   ALTERNATING   CURRENTS. 

132.  Electromotive  force  relations  of  the  rotary  converter. — Let 

E^  be  the  e.  m.  f.  between  the  direct- current  brushes  and  E    the 

effective  alternating  e.  m.  f.  between  adjacent  collecting  rings  of 

.    E 
an  ;z-ring  converter.     The  ratio  -~  has  a  characteristic  value  for 

each  value  of  n. 

Fundamental  assumption. — Consider  an  armature  conductor  c, 
Fig.  163,  at  angular  distance  /9  from  the  axis  of  the  field,  as 


Fig.  163. 

shown.*  We  assume  that  thee.  m.  f.  induced  in  the  conductor 
c  is  proportional  to  cos  /?  or  equal  to  C  cos  /9  where  C  is  a  con- 
stant. The  results  of  this  assumption  are  practically  in  accord 
with  experiment. 

The  number  of  armature  conductors  between  c  and  c'  is  pro- 
portional to,  or  say  equal  to,  d^. 

The  e.  m.  f.  in  each  conductor  is  C.  cos  ^,  and  the  e.  m.  f.  in 
all  the  conductors  between  c  and  c'  is  : 

de  =  C.  cos  /9.  d^.  .  (a) 

E.  m.  f.  E^  between  direct  cwrent  h'ushes. — All  the  conductors 
between  b  and  b'  are  in  series  between  the  direct-current  brushes 
so  that 

*The  discussion  in  Arts.  I32,  133  and  134  is  given  for  the  case  of  a  two-pole  ma- 
chine.    The  results,  however,  apply  to  multipolar  machines  as  well. 


THE   ROTARY   CONVERTER.  1/3 

X+90O 
C.  COS  3.  dB 
90O 

or£Q=2C.  (b) 

Effective  e.  m.f.  E^  between  adjacent  collecting  rings  of  an  n-ring 

converter. — ^The  e.  m.  f.  between  adjacent  rings  r  and  r' ,  Fig.  164, 

is  at  its  maximum  value  when  the  arc  rr'  is  bisected  by  the  axis 


Fig.  164. 

2.7t 

of  the  field  as  shown.     The  angle  between  r  and  r'  is  —  or 

n 

7t  _ 

half  this  angle  is  -  .     The  maximum  e.  m.  f.  ^2  E^  between  rings 
r  and  r'  is  therefore  : 

TT 

■s/2  E^=    \  ^      C.  COS  /?.  d^  =  2C  sin  - 
or  since  2C  =  E^we  have  . 

E^=  —^E, sin  Tt'in.  (96) 

V2 

Examples :    The  effective  alternating  e.  m.  f  of  a  single-phase 
converter  (?2  =  2)  is  : 

^2 =4^= -707  ^0-  (97) 

The  effective  alternating  e.  m.  f  between  adjacent  rings  of  a 
three-phase  converter  {11  =  3)  is  : 


174      THE   ELEMENTS    OF   ALTERNATING   CURRENTS. 


The  effective  alternating  e.  m.  f.  between  adjacent  rings  of  a 
two-phase  converter  (;?  =  4)  is  : 


^=5 
'       2' 


(99) 


Fig.   165. 


The  effective  e.  m.  f.  between  opposite  rings  of  a  two-phase 
converter  is  E^. 

133.  Current  relations  of  the  rotary  converter.  Fundamental 
assumptions. — In  the  discussions  of  current  relations  we  shall  as- 
sume that  the  alternating  current  flowing  through  each  section 

(between  adjacent  collecting  rings)  of 
the  armature  is  exactly  opposite  in 
phase  to  the  alternating  e.  m.  f.  in  that 
section,  and  that  the  intake  and  output 
of  power  are  equal. 

Let  /p  be  the  output  of  direct  cur- 
rent and  let  I^  be  the  effective 
alternating  current  flowing  in  the  armature  between  two  ad- 
jacent collecting  rings.  The  intake  of  power  per  phase  is  EJ^ 
or  the  total  intake  is  nEJ^  and  the  power  output  is  EJ^.  There- 
fore 

EI  —  nE  I 

or  substituting  the  value  of  E^  from  equation  (96)  we  have  : 

/     =    ; V-  (100) 

"      n  sin  7t\n 
Current  in  each  ^nain. — The  current  /  in  each  main  or  the 

T 

current  entering  the  armature  at  each  collecting  ring,  is  the  vector 
difference  between  I  J  and  I ^' ,  Fig.  165,  so  that 

/  =  2/  sin  Tzjn,  (loi) 

Examples :  The  effective  alternating  current  in  each  half  of 
the  armature  of  a  single-phase  converter  {it  =  2)  is : 


(102) 


THE   ROTARY   CONVERTER. 


175 


and  the  effective  alternating  current  entering  at  each  collecting 


ring  IS 


/  =  2/, 


1-414  4 


(103) 


The  effective  alternating  current  flowing  in  the  armature  be- 
tween adjacent  collecting  rings  of  a  three-phase  converter  {ii  =  3) 


IS  : 


1  =  ^1 


3^3 


(104) 


and  the  effective  current  entering  at  each  collecting  ring  is 

The  effective  alternating  current  flowing  in  the  armature  be 
tween  adjacent  collecting  rings  of  a  two-phase  converter  Qi  =  4) 

is:  ^4=K^  (106) 

and  the  effective  current  entering  at  each  collecting  ring  is 

/  =  ^/2/,.  (107) 

134.  Discussion  of  armature  heating  of  the  rotary  converter. — 

Let  rand  /,  Fig.    166,  be  the  points  of  attachment  of  adjacent 


Fig.   166. 


collecting  rings  of  an  ;2-ring  converter  and  let  the  line  6^ J/ bisect 
the  arc  r  r'.  Consider  an  armature  conductor  c  between  r  and  r' 
and  let  the  angle  cOM  be  represented  by  a.  The  largest  pos- 
sible value  of  a  is  71/71  or  one-half  of  the  angle  between  r  and  /. 


176      THE    ELEMENTS   OP    ALTERNATING  CURRENTS. 

Let  oit  be  the  angle  between  OM  and  the  axis  of  the  field.  The 
conductor  c  is  at  the  brush  h  when  wt  =  —  (90°  +  a),  and  at  the 
brush  b'  when  wt  =  90°  —  a.  During  this  time  half  of  the 
direct  current  /^  flows  through  the  conductor  c  in  an  unchanging 
direction. 

The  alternating  current  /  (effective  value)  flowing  between 
rings  r  and  r'  is  at  its  maximum  value  \^2  I ^  when  the  angle  mt. 
Fig.  166,  is  zero.*  This  alternating  current  is  therefore  equal  to 
\/2/^  cos  Oit  at  each  instant,  and  it  is  opposite  to  the  direct 
current  in  direction.  Therefore  the  total  current  in  the  conduc- 
tor c  during  the  time  from  iot  =  —  (90°  +  a)  to  oit  =  90°  —  «,  is: 

2  =  -^  —  '^2  I  cos  (tit, 

2 
or,  using  the  value  of  /  from  equation  (100),  we  have : 

2  \  n  sm  njit  /  ^       ^ 

The  average  square  of  this  current  during  the  time  oit=  —  (90° 
-f  a)  to  (i)t  =  90°  —  a  is  : 

4  \         7nis\n7il7i      71  sm.  Tzlnf 

and  the  heat  generated  in  the  conductor  c  is  proportional  to  this 
average  square.     The  heat  generated  in  the  conductor  by  the 

direct  current  alone  is  proportional  to  -5--  Therefore  the  conduc- 
tor c  has  I  I ; ^  +    -  .  „    ,   \t  times  as  much  heat  gen- 

\         nn  sm  tt/ ;?       n  sm  Tzjjt  J 

erated  in  it  when  the  machine  is  used  as  an  ?z-ring  converter  as 
is  generated  in  it  when  the  machine  gives  the  same  output  of 
direct  current  as  a  dynamo. 

*  In  and  En  assumed  to  be  exactly  opposite  in  phase  ;  see  Art.  133. 
f  Generally  less  than  unity. 


THE   ROTARY   CONVERTER.  177 

The  average  heating  over  the  entire  armature  is  found  by  in- 
tegrating  the  expression  (a)  with  respect  to  a  from  a  = to 

a=  +  -and  dividing;  the  result  by This  gives  : 

Average  heating  of  armature  of  -^  o    /  ^  ^  _l  ^  \  (\<\ 

;«-ring  converter  is  proportional  to  :  a    y  ;j-2        ^^2  gj^2  ^ /^^  ^  \    ) 

(t6  8  \  * 

71^  ^  ;/"  sin"  7:\n  ) 
times  as  great  as  the  heating  of  the  armature  by  the  direct  cur- 
rent alone.     Therefore  an  «-ring  converter  can  put  out 

I 


J 


i6 

!--,-  + 


71^        71^  sin^  7iliz 

times  as  much  direct  current  as  the  same  machine  can  when  used 
as  a  simple  dynamo,  for  the  same  total  armature  heating.  The 
table  given  in  Art.  131  is  calculated  in  this  way. 

Remark:  The  conductors  midway  between  the  points  of  at- 
tachment of  the  collecting  rings  (a  =  o)  are  heated  least,  and  the 
conductors  near  the  points  of  attachment  of  the  collecting  rings 

I  a  =  zt  —  J  are   heated  most.     Thus  in  a  two-ring    converter 

[n  =  2)  the  conductors  midway  between  the  points  of  attachment 
have,  according  to  the  expression  (a),  ^^-g-  as  much  heat  gener- 
ated in  them  as  would  be  generated  by  the  direct  current  alone  ; 
and  the  conductors  near  the  points  of  attachment  of  the  collecting 
rings  have  3  times  as  much  heat  generated  in  them  as  would  be 
generated  by  the  direct  current  alone. 


*  Generally  less  than  unity. 


CHAPTER  XIV. 


THE  INDUCTION  MOTOR. 


135.  The  utilization  of  alternating  current  for  motive  pur- 
poses.— It  has  been  already  pointed  out  that  the  successful  em- 
ployment of  alternating  current  for  motive  purposes  depends  upon 
the  use  of  the  mduction  motor  driven  by  polyphase  currents. 
The  synchronous  motor  (and  rotary  converter)  operates  satis- 
factorily with  single-phase  current  when  once  it  is  started,  but 
if  power  is  to  be  taken  directly  from  the  supply  mains  for  start- 
ing, then  polyphase  currents  are  most  satisfactory,  inasmuch  as 
the  synchronous  motor  may  then  be  started  by  an  auxiliary  in- 
duction motor.* 

136.  The  induction  motor  consists  of  a  primary  member  and  a 
secondary  member,  each  with  a  winding  of  wire.     The  primary 


Fig.  167.  Fig.  168. 

member  is  usually  stationary,  and  is  often  called  the  stator.    The 

*  The  polyphase  synchronous  motor  itself  acts  as  an  induction  motor  when  running 
below  synchronism,  especially  if  its  field  coils  are  short-circuited,  and  it  may,  there- 
fore, be  started  without  the  use  of  an  auxiliary  induction  motor.  This  method  of 
starting  is  used  by  the  General  Electric  Company. 

(178)    ' 


THE   INDUCTION   MOTOR. 


179 


secondary  member  is  usually  the  rotating  member,  and  is  often 
called  the  rotor.  Fig.  167  shows  a  rotor  of  the  squirrel  cage 
type.  It  consists  of  a  drum,  A,  built  up  of  circular  sheet-iron 
disks ;  near  the  periphery  of  this  drum  are  a  number  of  holes 
parallel  to  the  axis  of  the 
drum  ;  in  these  holes  heavy 
copper  rods,  b,  are  placed, 
and  the  projecting  ends  of 
these  rods  are  soldered  to 
massive  copper  rings,  r,  one 
at  each  end  of  the  drum. 
Another  type  of  rotor  is  de- 
scribed later. 

The  stator  is  a  laminated 
iron  ring,  FF,  Fig.  168, 
closely  surrounding  the  ro- 
tor. This  ring  is  slotted  on 
its  inner  face,  as  shown, 
windings  are  arranged  in  these  slots,  and  these  windings  receive 
currents  from  polyphase  supply  mains.  These  polyphase  cur- 
rents produce  in  the  stator  a  rotating  state  of  magnetism,  the 
action  of  which  on  the  rotor  is  the  same  as  the  action  of  an  ordi- 
nary field  magnet  in  rotation.  Thus,  Fig.  169  shows  a  squirrel 
cage  rotor.  A,  surrounded  by  an  ordinary  field  magnet  rotating 
in  the  direction  of  the  curved  arrows.  This  motion  of  the  field 
magnet  induces  currents  in  the  short-circuited  copper  rods  of  the 
rotor ;  the  field  magnet  exerts  a  dragging  force  on  these  currents 
and  causes  the  rotor  to  rotate. 

No  electrical  connections  of  any  kind  are  made  to  the  rotor. 
The  next  article  describes  the  stator  windings  and  explains  the 
manner  in  which  these  windings  produce  the  rotating  state  of 
magnetism  in  the  stator. 

137.  Stator  windings  and  their  action. — The  stator  windings 
are  arranged  in  the  slots  s,  Fig.  168,  in  a  manner  exactly  similar 


Fig.  169. 


l8o   THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 


to  the  arrangement  of  the  windings  of  the  two-phase  or  three- 
phase  alternator  armature,  according  as  the  motor  is  to  be  sup- 
pHed  with  two  or  three-phase  currents. 

Fig.  170  shows  an  end  view  of  a  four-pole  two-phase  induction 
motor.  In  this  figure  the  outline,  only,  of  the  rotor  is  shown ; 
the  stator  conductors  are  represented  in  section  by  the  small 
circles  ;  the  slots  are  omitted  for  the  sake  of  clearness  ;   and  the 

end  connections  of  half 
the  stator  conductors 
are  shown  in  Fig.  171. 
The  stator  conductors 
are  arranged  in  two 
distinct  circuits.  One 
of  these  circuits  includes 
all  of  the  conductors 
marked  A  and  receives 
current  from  one  phase 
of  a  two -phase  system  ; 
the  other  circuit  in- 
cludes all  of  the  con- 
ductors marked  B  and 
receives  current  from 
the  other  phase  of  the  two-phase  system.  The  terminals  of  the 
B  circuit  are  shown  at  tt',  Fig.  171.  The  conductors  which 
constitute  one  circuit  are  so  connected  that  the  current  flows  in 
opposite  directions  in  adjacent  groups  of  conductors  as  indicated 
by  the  arrows  in  Fig.  171.  The  radial  lines  in  Fig.  171  rep- 
resent the  stator  conductors  and  the  curved  lines  represent  the 
end  connections,  as  in  the  winding  diagrams.  Figs.  98  to  105. 

The  action  of  a  band  of  conductors  between  two  masses  of 
iron  is  shown  in  Figs.  172  and  173.  The  small  circles  in  these 
figures  represent  the  conductors  in  section  ;  conductors  carrying 
down-flowing  currents  are  marked  with  crosses,  those  carrying 
up-flowing  currents  are  marked  with  dots,  and  those  carrying  no 


Fig.   170. 


THE   INDUCTION   MOTOR. 


i8i 


current  are  left  blank.  The  action  of  the  currents  in  these  bands 
of  conductors  is  to  produce  magnetic  flux  along  the  dotted  lines 
in  the  directions  of  the  arrows. 

The  lines  A'  and  B'  in  Figs.  174,  175  and  176  are  supposed 


Fig.  171. 

to  rotate  and  their  projections  on  the  fixed  line  ef  represent  the 
instantaneous  values  of  the  alternating  currents  in  the  A  and  B 
conductors  respectively. 


iron 


!  OOQQG: 


iron 


iron'  I 

UV     flowinq  current^ 

Fig.  172. 

Fig.  1 74  shows  the  state  of  affairs  when  the  current  in  conduc- 
tors A  is  z.  maximum  and  the  current  in  conductors  B  is  zero. 


I — '>..-: — ^' 

\  iron 

down    flowina-  current^ 
Fig.  173. 


1 82       THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 

The  dotted  lines  indicate  the  trend  of  the  magnetic  flux.  This 
flux  enters  the  rotor  from  the  stator  at  the  points  marked  N  and 
leaves  the  rotor  at  the  points  m.arked  S. 


Fig.  174. 


Fig.  175. 


THE   INDUCTION   MOTOR. 


183 


Fig.  175  shows  the  state  of  affairs,  ^  of  a  cycle  later,  when 
the  current  in  the  B  conductors  has  increased  and  the  current  in 
the  A  conductors  has  decreased  to  the  same  value.  The  points 
N  and  vS  have  moved  over  Jg  of  the  circumference  of  the  stator 
ring. 

Fig.  176  shows  the  state  of  affairs,  after  another  eighth  of  a 
cycle,  when  the    current  in  the   B  conductors  has  reached  its 


maximum  value  and  the  current  in  the  A  conductors  has  dropped 
to  zero.  The  points  N  and  5"  have  moved  again  over  Jg-  of  the 
circumference  of  the  stator  ring. 

This  motion  of  the  points  N  and  6"  is  continuous,  and  these 
points  make  one  complete  revolution  (in  a  four-pole  motor)  dur- 
ing two  complete  revolutions  of  the  vectors  A'  and  B'  or  while 
the  alternating  currents  supplied  to  the  stator  windings  are  pass- 
ing through  two  cycles.     In  general 

/ 


n  = 


in  which  n  is  the  revolutions  per  second  of  the  stator-magne- 


1 84       THE   ELEMENTS    OF   ALTERNATING   CURRENTS. 

tism,  p  is  the  number  of  pairs  of  poles  N  and  vS",  and  f  is  the  fre- 
quency of  the  alternating  currents  supplied. 

138.  Preliminary  discussion  of  the  action  of  the  induction  motor. 

— The  complete  theoretical  discussion  of  the  action  of  the  induc- 
tion motor  is  given  later  and  is  in  many  respects  similar  to  the 
theory  of  the  transformer.  Many  important  details  of  the  action 
of  the  induction  motor,  however,  are  most  easily  explained  by 
looking  upon  the  induction  motor  as  a  rotor  influenced  by  a  ro- 
tating field  magnet. 

Torque  and  speed, — Let  n  be  the  revolutions  per  second  of  the 
field  and  n'  the  revolutions  per  second  of  the  rotor.  When  n  =  n' 
the  rotor  and  field  turn  at  the  same  speed,  so  that  their  relative 
motion  is  zero  ;  no  e.  m.  f  is  then  induced  in  the  rotor  conductors 
and  no  current,  and  therefore  the  rotating  field  exerts  no  torque 
upon  the  rotor.  As  the  speed  of  the  rotor  decreases  the  relative 
speed  of  rotor  and  field  increases,  and  therefore  the  e.  m.  f  induced 
in  the  rotor  conductors,  the  currents  in  the  conductors,  and  the 
torque  with  which  the  field  drags  the  rotor,  all  increase.  If  the 
whole  of  the  field  flux  were  to  pass  into  the  rotor  and  out  again 
in  spite  of  the  demagnetizing  action  of  the  currents  in  the  rotor 
conductors,  then  the  torque  would  increase  in  strict  proportion  to 
n  —  n' ^  but  in  fact  a  larger  and  larger  portion  of  the  field  flux 
passes  through  the  space  between  stator  and  rotor  conductors  as 
the  speed  of  the  rotor  decreases  and  this  magnetic  leakage  causes 
the  torque  to  increase  more  and  more  slowly  as  n  —  n'  increases, 
only  in  some  cases*  reaching  a  maximum  value  and  then  decreas- 
ing with  further  incease  of  n  —  n' . 

Fig.  I  "JJ  shows  the  typical  relation  between  torque  and  speed 
of  an  induction  motor.  Ordinates  of  the  curve  represent  torque 
and  abscissas  measured  from  0  represent  rotor  speeds.  The  rotor 
is  said  to  run  above  synchronism  when  it  is  driven  so  that  n'  >  n. 

Use  of  starting  resistance  in  the  rotor  windings. — The  speed  of 

*  In  every  case,  if  one  makes  n  —  n'  large  enough  by  driving  tlie  rotor  backwarcis 

so  that  n'  becomes  negative. 


THE    INDUCTION    MOTOR. 


185 


rotor  for  which  the  maximum  torque  occurs  depends  upon  the 
resistance  of  the  rotor  windings,  and  it  is  advantageous  to  pro- 
vide at  starting  such  resistance  in  these  windings  as  to  produce 
the  maximum  torque  at  once,  this  resistance  being  cut  out  as  the 
motor  approaches  full  speed. 

Efficiency  and  speed. — Let  T  be  the  torque  with  which  the  ro- 
tating field  drags  on  the  rotor  ;  then  27171' T  is  the  power  taken  up 
by  the  rotor  to  be  givea  off  its  belt  pulley.  Also  T\s  the  react- 
ing torque  which  opposes  the  rotation  of  the  field,  so  that  27i:nT 


Fig.  177. 


is  the  power  required  to  maintain  the  rotation  of  the  field.  There- 
fore, ignoring  friction  losses,  2T[nTis  the  power  intake  and  27zn'T 
is  the  power  output  of  the  motor,  so  that : 

27tn'  T      n' 

:  (109) 


^  = 


2'KnT 


is  the  efficiency  of  the  machine.     This  equation  shows  that  the 

efficiency  of  an  induction  motor  is  zero  when  the  rotor  stands 

still,  that  it  increases  as  the  rotor  speeds  up,  and  approaches  100^ 

(ignoring  field  losses  and  friction)  as  the  rotor  speed  approaches 

n' 
the  field  speed.     The  ratio  —  ranges  from  .85  to  .95  or  more  in 

commercial  induction  motors  under  full  load. 

Efficiency  and  rotor  resistance. — For  a  given  difference   n  —  n' 
between  field  speed  and  rotor  speed,  given  e.  m.  f  is  induced  in 


1 86       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

the  rotor  conductors,  and  the  less  the  rotor  resistance  the  greater 
the  current  produced  by  this  e.  m.  f.  and  the  greater  the  torque. 
Therefore  a  given  induction  motor  will  develop  its  full  load  torque 

for  a  small  value  of  n  —  n'  or  for  a  larger  value  of  —  (efhciency) 

n 

if  its  rotor  resistance  is  small.  High  efficiency  depends,  there- 
fore, upon  low  rotor  resistance. 

The  induction  generator. — When  the  rotor  of  an  induction 
motor  is  driven  above  synchronism  (?/  >• ;?),  by  an  engine  for 
example,  the  torque  is  reversed  and  opposes  the  motion  of  the 
rotor  so  that  2nriT  is  input  and  27znT  is  output.  That  is,  the 
machine  takes  power  from  the  engine  to  drive  its  rotor  and  gives 
out  power  from  its  stator  windings.  This  output  of  power  is  in 
the  form  of  polyphase  currents  the  frequency  of  which  is  fixed  by 
the  frequency  of  the  alternator  (or  synchronous  motor)  which  is 
connected  to  the  stator  windings. 

139.  The  driving'  of  induction  motors  by  single-phase  alternat- 
ing current. — This  is  accomplished  by  connecting  the  two  stator 
circuits  A  and  B  (case  of  two-phase  motor)  in  parallel  to  the 
single-phase  supply  mains  at  the  same  time  connecting  in  series 
with  A  a  resistance  R  (see  Fig.  178).  The  currents  in  the  cir- 
cuits A  and  B  then  differ  in  phase 
sapphi  main  on  account  of  the  dissimilarity  of 

-^— ^         the  circuits,  and  the  motor  starts. 
_,     When  the   motor   is    well    under 
way  one  of  the  windings  A  or  B 
j^\        *-r         may  be  short-circuited,  the   other 
5*    ■     I  only  being  left  connected  to  the 

aappli/main  \  •>  ^ 

'  '  „  mains.     A  two -phase,   or  even  a 

Fig.  178.  ^ 

three-phase  motor  operates  fairly 
well  under  these  conditions,  except  that  excessive  current  is  re- 
quired at  starting  to  give  a  good  starting  torque.  The  resistance 
R  may  be  replaced  with  advantage  by  a  condenser,  especially 
i.i  case  of  a  small  motor. 


THE   INDUCTION   MOTOR.  18/ 

Remark  :  The  foregoing  discussion  refers  explicitly  to  the  two 
phase  motor.  The  three-phase  motor  differs  but  little  from  the 
two-phase  motor,  as  will  appear  in  the  following  discussion. 

General  Theory  of  the  Induction  Motor. 

140.  The  general  alternating  current  transformer. — The  general 
theory  of  the  induction  motor  is  best  developed  by  considering 
at  once  the  most  general  type  of  machine,  a  multipolar  multi- 
phase motor,  of  which  the  rotor  is  wound  in  precisely  the  same 
way  as  the  stator,  the  rotor  windings  being  connected  to  collect- 
ing rings,  so  that  the  currents  induced  in  the  rotor  windings  may 
be  available  for  outside  purposes.  Such  a  machine  we  will  call 
the  general  alternating  current  transformer.  Thus,  a  2/'-pole, 
^-phase  machine  would  have  its  stator  conductors  arranged  in 
q  distinct  circuits,  each  taking  current  from  one  phase  of  a  ^-phase 
system  ;  furthermore,  each  circuit  would  include  2p  equidistant 
groups  of  conductors  so  connected  that  a  current  in  that  circuit 
would  flow  in  opposite  directions  in  adjacent  groups.  The  rotor 
conductors  would  be  similarly  arranged  in  ^  *  distinct  circuits, 
each  connected  to  a  pair  of  collecting  rings  and  supplying  cur- 
rent to  an  outside  receiving  circuit. 

Of  course,  no  such  induction  motors  f  are  ever  actually  built, 
but  it  is  important  to  have  clearly  in  mind  the  details  of  the  ma- 
chine to  which  the  following  discussion  applies. 

Fig.  179  shows  a  little  more  than  one-sixth  part  of  the  circum- 
ference of  a  six-pole,  three-phase  machine.     The  three  groups  of 

*  Stator  and  rotor  are  not  necessarily  wound  for  the  same  number  of  phases,  but 
the  discussion  is  simpUfied  by  such  an  arrangement. 

•f  Steinmetz  has  proposed  the  use  of  such  induction  motors  for  street  railway  work. 
Two  similar  motors  are  used  on  each  car,  one  geared  to  each  axle.  At  starting  and 
for  slow  running  motor  No.  I  takes  currents  into  its  stator  windings  from  polyphase 
trolley  wires,  and  supplies  polyphase  currents  from  its  rotor  windings  to  the  stator  of 
motor  No.  2,  the  starting  resistance  being  connected  in  the  rotor  circuits  of  motor 
No.  2.  With  such  an  arrangement  the  limit  of  speed  is  one-half  of  synchronous 
speed  [n^=)4  n),  and  the  efficiency  at  given  speed  is  doubled.  For  fast  running 
both  motors  take  current  directly  from  the  trolley  wires. 


1 88       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


stator  conductors,  A,  B  and  C,  belong  one  to  each  of  the  three 

circuits  formed  by  the  stator  windings ;  and  the  three  groups  of 

rotor  conductors,  A' ,  B'  and  C ,  belong  one  to  each  of  the  three 

circuits  formed  by  the  rotor  windings. 

When  the  rotor  of  such  a  machine  is  stationary  the  machine 

acts  simply  as  a  transformer  taking  ^-phase  currents  from  the 

supply  mains  into  its  stator  windings 
and  giving  out  ^'-phase  currents  of 
the  same  frequency  from  its  rotor 
windings.  If  stator  and  rotor  have 
the  same  number  of  conductors, 
which  is  understood  to  be  the  case 
in  the  following  discussion,  then  the 
ratio  of  transformation  is  i  :  i  when 
the  rotor  is  stationary. 

141.  Effect  of  rotor  speed  upon  the 
transformer  action  of  the  induction 
motor. — In  the  discussion  of  this  mat- 
ter it  is  convenient  to  express  speed 
of  rotor  and  speed  of  stator-magnet- 
ism  not  in  revolutions  per  second  but 
in  terms  of  what  we  may  call  briefly 
rotation  frequency  which  is  the  pro- 
duct of  revolutions  per  second  into 

the  number  of  pairs  of  poles  for  which  the  machine  is  wound. 

The  rotation  frequency  of  stator  magnetism  is  the  frequency  of 

the  polyphase  currents  supplied  to  the  stator  windings. 

Let  f  be  the  rotation  frequency  of  stator-magnetism,  and  /' 

the  rotation  frequency  of  the  rotor.     The  relative  speed  of  the 

two  is  then/—/'. 

When  the  rotor  is  stationary  (/'  =  o)  the  relative  speed  of 

rotor  and  stator-magnetism  is  /  and  the  e.  m.  f  induced  in  each 

rotor  circuit  is  of  the  same  value  and  frequency  as  the  e.  m.  f. 

acting  on  each  circuit  of  the  stator  winding  as  pointed  out  in  the 

foregoing  article. 


Fig.  179. 


THE   INDUCTION   MOTOR.  1 89 

When  the  rotor  speeds  up  to  a  rotation  frequency  of  f  the 
relative  speed  of  rotor  and  stator-magnetism  is  reduced  to  f—f, 
and  the  e.  m.  f.  induced  in  each  rotor  circuit  is  reduced  in  value 
and  in  frequency  in  proportion  to  the  decrease  in  relative  speed  ; 

/-/ 
that  is  in  the  ratio  of  /:  f—f .     This  ratio  — 7—  is  called  the 

slip,  s  of  the  machine.  Therefore  when  the  rotation  frequency 
of  the  rotor  is  /'  the  e.  m.  f  induced  in  each  rotor  circuit  is  sE^ 
and  its  frequency  is  sf ;  where  E^  is  the  value  of  the  e.  m.  f.  act- 
ing on  each  circuit  of  the  stator  winding,  stator  and  rotor  con- 
ductors being  equal  in  number. 

The  frequency  changer. — The  induction  motor  is  sometimes 
used  as  a  so-called  frequency  changer.  Alternating  currents  at 
a  given  frequency  are  taken  into  the  stator  windings.  The  rotor 
is  loaded  by  belt  to  bring  it  to  such  speed  that  currents  of  re- 
quired frequency  may  be  taken  from  the  rotor  windings. 

142.  The  use  of  a  fictitious  frequency  for  the  alternating  currents 
in  the  rotor  windings. — Neither  the  graphical  method  nor  the  sym- 
bolic method  can  be  satisfactorily  used  in  the  discussion  of  an  al- 
ternating current  problem  in  which  it  is  necessary  to  consider 
alternating  currents  of  different  frequency  simultaneously,  such 
as  the  alternating  currents  in  the  stator  and  rotor  windings  of  the 
induction  motor. 

The  e.  m.  f  induced  in  a  given  rotor  conductor  and  the 
current  flowing  in  the  conductor  actually  pass  through  f—f 
cycles  per  second.  Let  us  consider  however,  not  the  suc- 
cessive instantaneous  values  of  e.  m.  f  and  current  in  a  given 
rotor  conductor,  but  the  instantaneous  values  of  e.  m.  f  and 
current  in  the  successive  rotor  conductors  as  they  pass  a  given 
stator  conductor.  These  e.  m.  f  and  current  values  pass  through 
/  cycles  per  second,  their  maximum  (and  effective)  values  are 
the  same  as  the  maximum  (and  effective)  values  of  e.  m.  f  and 
current  in  a  given  rotor  conductor,  and  their  average  prod- 
uct gives  the  average  power  developed    in  a  given   rotor  con- 


igo      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


ductor.*  By  employing  this  fictitious  frequency  the  electric  and 
magnetic  actions  of  the  induction  motor  become  identical  to  those 
of  the  simple  transformer.  The  primary  and  secondary  e.  m.  f 's 
(each  phase)  are  ^^  and  s£^  and  the  primary  and  secondary  currents 
are  equal  and  opposed  to  each  other  in  their  magnetizing  action. 
143.  The  ideal  induction  motor. — An  induction  motor  of  which 
stator  and  rotorf  windings  have  negligible  resistance,  of  which  the 
magnetic  circuits  have  negligible  reluctance,  in  which  hysteresis 
and  eddy  currents  are  negligible,  and  which  satisfies  the  condition 
that  all  the  magnetic  flux  through  the  stator  passes  also  through 
the  rotor  is  called  an  ideal  induction  motor. 

Let  E^,  Fig.  1 80,  represent  the  e.  m.  f  acting  on  one  circuit  of 
the  stator  winding.  Then  E^i—  ^-^1)  '^^  ^^^  ^-  ^-  ^-  induced  in  one 
of  the  rotor  circuits.  The  current  I^  which  the  e.  m.  f  E^  pro- 
duces is  determined  by  the  resistance  and  induc- 
tance of  the  outside  circuit  to  which  it  supplies 
current,  according  to  Problem  IV.,  Chapter  V. 
The  current  /  ^  in  the  given  stator  circuit  is  equal 
to  /g.  The  input  of  power  per  phase  (z.  c,  the 
power  put  into  one  stator  circuit)  is  E^I^  cos  d ; 
the  output  of  electrical  power  per  phase  is  E^/^ 
cos  6  or  sE^I^  cos  d  since  E^  =  sE^  and  I^  —  I^; 
and  the  output  of  mechanical  power  per  phase 
is  the  difference  between  input  and  electrical 
output  or  (i  —  s)  E,J^  cos  d. 

Example  :  A  four-pole,  three-phase  machine 
takes  current  at  220  volts  (each  phase),  the  fre- 
quency being  60  cycles  per  second.  When  the 
rotor  is  stationary  the  rotor  gives  out  three-phase  currents  at 
220  volts  and  full  frequency.  When /"'=  30  (rotor  speed  15 
revolutions   per    second)  the    rotor  gives    out  three-phase  cur- 

*  Strictly-j  the  average  power  developed  in  all  the  rotor  conductors  while  they  are 
passing  a  given  stator  conductor — which  amounts  to  the  same  thing. 

f  The  resistance  of  the  rotor  windings  may  be,  and  is  hereafter,  included  with  the 
resistances  of  the  external  circuits  receiving  currents  from  the  rotor. 


Fig.  180. 


THE   INDUCTION   MOTOR.  I9I 

rents  at  1 10  volts  and  half  frequency,  and  of  the  total  power  in- 
take half  is  given  out  as  mechanical  energy  and  half  as  electrical 
energy.  When/"' =  50  (rotor  speed  25  revolutions  per  second) 
the  rotor  gives  out  three-phase  currents  at  1  full  e.  m.  f.  and  fre- 
quency, and  of  the  total  power  intake  |  is  given  out  as  mechan- 
ical energy  and  |  as  electrical  energy. 

When/'  approaches  60  per  second  the  e.  m.  f.  induced  in  each 
rotor  circuit  approaches  zero  frequency  and  zero  value,  and  the 
total  intake  falls  off,  but  a  larger  and  larger  portion  of  the  intake 
appears  as  output  of  mechanical  energy.  In  this  example  the 
resistance  of  each  circuit  receiving  current  from  rotor  windings  is 
supposed  to  be  constant. 

Remark :  The  action  of  the  actual  induction  motor  deviates 
from  the  above  described  ideal  action  because  of  the  resistance 
of  the  stator  windings,  because  of  magnetic  reluctance,  eddy  cur- 
rents and  hysteresis,  and  because  of  magnetic  leakage.  These 
things  are  very  nearly  independent  of  each  other  in  their  action, 
and  therefore  the  effect  of  each  will  be  considered  by  itself. 
Further,  the  discussion  of  the  eiTects  of  these  things  is  almost 
identical  to  the  discussion  of  their  effects  on  the  simple  transfor- 
mer ;  therefore,  those  things  only  will  be  fully  discussed  which 
have  a  bearing  upon  the  motor  action  of  the  machine,  that  is, 
which  have  influence  upon  the  relation  between  torque  and  speed. 

Magnetic  leakage  and  rotor  resistance  (including  resistance  of 
entire  secondary  circuits)  have  very  great  influence  upon  the  be- 
havior of  the  machine  as  a  motor  ;  that  is,  upon  speed  and  torque. 

Stator  resistance,  magnetic  reluctance,  eddy  currents  and  hys- 
teresis are,  under  practical  conditions,  almost  without  influence 
upon  speed  and  torque,  their  effect  being  mainly  to  cause  the 
stator  to  take  from  the  mains  slightly  more  current  and  slightly 
more  power  for  a  given  rotor  output  than  would  be  the  case  with 
an  ideal  machine. 

144.  Effect  of  magnetic  reluctance,  eddy  currents  and  hysteresis 
upon  the    action  of  an   induction  motor. — The  ideal    induction 


192       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


motor  takes  no  current  from  the  mains  into  its  stator  windings 
when  the  motor  is  running  at  synchronous  speed  (/'  =/).  The 
actual  induction  motor  running  in  synchronism  takes  sufficient 
current  to  overcome  the  magnetic  reluctance  of  the  iron  (stator 
and  rotor);  and  an  amount  of  power  equal  to  the  hysteresis  and 
eddy  current  loss  in  the  stator  iron,  only,  inasmuch  as  the  mag- 
netic state  of  the  rotor  is  constant,  since  it  rotates  with  the  stator 
magnetism.  When  the  actual  induction  motor  is  running  at  any 
given  speed  it  takes  from  the  mains  the  above  current  and  power 
in  excess  of  what  an  ideal  motor  would  take  at  same  speed. 
Further,  there  is  eddy  current  and  hysteresis 
loss  in  the  rotor  iron  when  its  runs  below 
synchronism  and  the  effect  of  this  loss  is  to 
slightly  increase  the  torque. 

145.  Effect  of  stator  resistance  upon  the 
action  of  an  induction  motor. — When  the  ro- 
tor is  running  nearly  in  synchronism  with 
the  stator  magnetism  the  currents  in  the  sta- 
tor windings  are  very  small  and  no  perceptible 
portion  of  the  supply  e.  m.  f.  is  needed  to 
overcome  the  stator  resistance.  As  the  rotor 
is  slowed  up  the  stator  currents  increase  and 
a  larger  and  larger  portion  of  the  supply  e. 
m.  f.  is  needed  to  overcome  the  stator  resis- 
tance. The  result  is  that  the  core  flux*  falls 
off  slightly,  and  also  the  torque  acting  upon  the  rotor,  inasmuch 
as  this  torque  depends  upon  both  flux  and  rotor  currents. 

146.  Effect  of  magnetic  leakage  upon  the  action  of  an  induction 
motor. — As  in  case  of  the  simple  transformer  the  effect  of  mag- 
netic leakage  is  the  same  as  the  effect  of  an  outside  inductance 
connected  in  series  with  the  primary  (stator)  windings,  a  separate 


Fig.  i8i. 


*  Inasmuch  as  the  portion  of  the  supply  e.  m.  f.  which  is  balanced  by  the  induced 
e.  m.  f.  in  the  stator  windings  is  decreased,  and,  therefore,  the  harmonically  varying 
flux   which  induces  this  e.  m.  f.  must  decrease  exactly  as  in  the  simple  transformer. 


THE    INDUCTION    MOTOR.  1 93 

inductance  for  each  stator  circuit.  Let  P  be  the  value  of  each 
inductance  and  let  x  (=  a>P)  be  its  reactance  value. 

The  diagram  of  Fig.  i8i  represents  the  action  of  an  induction 
motor  in  so  far  as  it  is  affected  by  magnetic  leakage ;  A  and  E^ 
(=  sA)  are  the  e.  m.  f 's  induced  in  stator  and  rotor  windings  re- 
spectively by  the  magnetic  flux  which  passes  through  both. 
The  current  I^  is  determined  by  the  resistance  and  reactance*  of 
the  secondary  circuit.  The  primary  current  /^  is  equal  and  op- 
posite to  /g.  The  line  xl^  at  right  angles  to  /^  represents  that 
part  of  the  total  primary  e.  m.  f  which  is  used  to  overcome  the 
leakage  inductance  P  or  to  balance  the  e.  m.  f  induced  in  each 
primary  circuit  by  the  leakage  flux. 

Let  R  be  the  resistance  of  each  rotor  circuit  (inductance  negli- 
gible). 

Since  I ^  and  I ^  are  equal  we  may  represent  both  by  the  sym- 

E 
bol  /.     The  secondary  current  is  -~  or  since  E^  =  sA  we  have  : 

1=  ^-  (a) 

When  the  angle  6  is  zero  then  xl^,  Fig.  i8i,  is  at  right  angles 
to  A  and  we  have 

E^^  =  A^  +  x^P 

or  using  the  value  of  /  from  equation  (a)  and  solving  for  A?  we 

E'^F? 
get  ^'  =  ^^tA^-  C^) 

The  power  intake  of  primary,  per  circuit,  is  P  =  AI  since  A 
is  the  component  of  E^  parallel  to  /.  Therefore  substituting  for 
/  its  value  from  (a)  and  for  A^  its  value  from  (b)  we  have  : 

pi  =,         1^ (c) 

*  Under  practical  conditions  the  rotor  circuits  are  noninductive  and  the  angle  6, 
Fig.  1 8 1,  is  zero,  as  in  case  of  the  simple  transformer  feeding  a  noninductive  receiving 
circuit. 

13 


194       THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 

The  electrical  power  output  per  secondary  circuit  is  P"  =E^I. 
Therefore  substituting  sA  for  E^,  substituting  for  /its  value  from 
(a),  and  substituting  for  A^  its  value  from  (b)  we  have  : 

R'  +  s'x"  ^^ 

The  mechanical  power  output  per  phase  is   M=  P' —P"  or : 

s  E'^R 

The  torque,  T,  acting  upon  the  rotor,  per  phase,  is  such  that 
M=27in'T.  (f) 

in  which  7t'  is  the  rotor  speed. 

Let/  be  the  frequency  of  the  supply  currents,  p  the  number 
of  pairs  of  poles  for  which  the  motor  is  wound,  and  n  the  speed 
of  the  stator-magnetism,  then 

f=pn.  (g) 

= I  be  the  slip,  then    n'  =  (i— ^)  n,  or  using 

the  value  of  n  from  equation  (g),  we  have 

n'=-^{i-s).  (h) 

Substituting  the  value  of  M  from  (e)  and  the  value  of  n'  from 
(h),  in  equation  (f )  we  have 


T  = 


p         sE^R 


T  =■  ■ J (110) 


2Tlf       R?-\-  S^X^ 

This  is  the  torque  per  phase  and  since  there  are  q  phases  the 
total  torque  is  : 

pq         sE^^R 

2^  ■  R^  +  s'x^ 

This  is  the  equation  to  the  curve,  Fig.  182,  of  which  the  ordi- 
nates  represent  the  torque  T  and  the  abscissas  represent  the  slip 
s  positive  to  the  left  and  negative  to  the  right.  This  figure  is 
essentially  the  same  as  Fig.  177. 


THE   INDUCTION   MOTOR. 


195 


Maximum  torque. — The   slip   s   corresponding   to   maximum 
torque  is  found  from  the  condition 

dT 


ds 


=  o 


which  gives 


s  =  ± 


R 


(III) 


The  value  of  the  maximum  torque  is  found  by  substituting 
this  value  o{ s  in  equation  (no)  which  gives  : 


Fig,  182, 

j^    _pg_  El 


(112) 


The  maximum  torque  is  therefore  independent  of  secondary 
resistance. 

Starting  torque. — When  .y  =  i  the  rotor  is  stationary  and  the 
corresponding  value  of  T  is  the  starting  torque  T^,  namely : 

pq     E^R 


■^1       27r/  R^^x"- 


(113) 


To  give  the  greatest  starting  torque  R  must  be  adjusted  to 
make  T^  a  maximum.     The  condition  is 


196       THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 


dR 
which  gives  R  =  x.  (^^3) 

That  is,  to  give  maJcimum  torque  at  starting  the  rotor  resist- 
ance per  circuit  must  be  equal  to  the  leakage  reactance  per 
circuit. 

147.  Calculation  of  leakage  reactance. — The  leakage  reactance 
X  per  circuit  is  equal  to  wP  (^  27ifP)  where  P  is  the  leakage  in- 
ductance per  circuit.  This  leakage  inductance  is  calculated,  as 
in  case  of  the  simple  transformer,  by  equation  {7S),  namely 


(y+Y+.^).  (75)  bis 


/       \3 

This  equation  gives  P'm  centimeters,  all  dimensions  being  in 
centimeters.  In  this  equation  X  in  the  length,  parallel  to  the 
shaft,  of  the  rotor  or  stator ;  /  is  the  sum  of  the  widths  of  all  the 
slots  in  which  the  windings  of  one  stator  circuit  are  wound  ;  X  is 
the  depth  of  the  stator  slots  ;  Fis  the  depth  of  the  rotor  slots  ; 
and  g  is  the  clearance  space  between  stator  and  rotor.  This 
equation  assumes  that  stator  and  rotor  slots  are  of  the  same 
width,  that  they  are  wound  full  of  wire,  and  that  the  permeability 
of  the  iron  lugs  between  the  slots  is  very  great  so  that  reluctance 
of  iron  is  negligible. 

148.  Outline  of  the  design  of  an  induction  motor.  Design  of 
primary. — This  member,  which  is  usually  the  stator  or  stationary 
member  in  practice,  is  designed  in  a  manner  precisely  similar  to 
the  design  of  a  polyphase  alternator  armature  as  follows  : 

Value  and  frequency  of  supply  e.  m.  f.,  speed  of  motor*  and 
output  are  usually  prescribed. 

Number  of  poles  follows  at  once  from  frequency  and  speed. 
Inner  diameter  of  stator  is  fixed  by  speed  of  rotation  and  allow- 
able peripheral  speed.  Length  is  then  determined  so  as  to  radi- 
ate the  internal  losses  and  so  on.     In  the  calculation  of  flux, 

*  Which  is  practically  the  speed  of  the  stator-magaetism. 


THE   INDUCTION   MOTOR.  197 

—  of  jthe  inner  face  of  the  primary  member  may  be  taken  as  the 

AP 

approximate  area  of  pole  face/  being  the  number  of  pairs  of  poles. 

Windings  are  usually  distributed  in  from  2  to  6  slots  per  pole 

per  phase.     The  relation  between  primary  e.  m.  f  per  phase, 

turns  T  per  phase,  flux  from  one  pole,  and  frequency  is  given  by 

equation  (59). 

Design  of  secondary. — Length  and  diameter  of  secondary  mem- 
ber is  determined  by  length  and  diameter  of  primary  member. 
The  secondary  should  have  about  the  same  amount  of  copper  as 
primary.  Squirrel  cage  winding  or  ordinary  distributed  winding 
may  be  used.  The  latter  is  frequently  employed  when  it  is  de- 
sired to  insert  resistance  in  rotor  winding  at  starting.  Number  of 
slots  in  secondary  is  usually  different  from  number  in  primary  to 
avoid  simultaneous  coincidence  of  all  primary  and  secondary  slots. 

149.  The  action  of  the  polyphase  alternator  as  an  induction 
motor  when  beings  started  as  a  synchronous  motor. — The  winding 
of  the  polyphase  armature  is  identical  to  the  stator  or  primary 
winding  of  an  induction  motor.  When,  at  starting,  the  armature 
is  connected  to  the  polyphase  supply  mains  a  rotating  state  of 
magnetism  is  set  up  in  the  armature  core.  This  rotating  magne- 
tism exerts  a  dragging  torque  on  the  field  magnet,  especially  if 
the  field  coils  are  short-circuited,  and  the  reacting  torque  of  the 
field  upon  the  armature  sets  the  latter  rotating  in  a  direction  op- 
posite to  the  direction  of  its  rotating  magnetism. 


CHAPTER   XV. 

TRANSMISSION   LINES. 

150.  Introductory. — Power  may  be  transmitted  by  the  pump- 
ing of  water.  If  great  pressure  is  used  a  given  amount  of  power 
may  be  transmitted  by  a  small  flow  of  water  through  a  small 
pipe.  In  every  case,  however,  there  is  a  loss  of  power  on  ac- 
count of  friction  in  the  pipe.  The  smaller  the  pipe  the  greater 
this  loss  and  the  less  the  first  cost ;  the  best  size  of  pipe  is  that 
for  which  neither  the  first  cost  nor  the  continuous  loss  of  power 
by  friction  is  excessive. 

Similarly  a  given  amount  of  power  may  be  transmitted  by  a 
small  electric  current  through  a  small  wire  by  using  a  large 
electrical  pressure  or  e.  m.  f  In  every  case,  however,  there  is  a 
loss  of  power  on  account  of  the  resistance  of  the  wire.  The 
smaller  the  wire  the  greater  this  loss  and  the  less  the  first  cost  of 
the  line  ;  the  best  size  of  wire  is  that  for  which  neither  the  first 
cost  nor  the  continuous  loss  of  power  by  resistance  is  excessive. 

It  is  only  by  using  very  large  e.  m.  f 's  that  long  distance  trans- 
mission lines  may  be  made  at  a  reasonable  cost,  the  loss  due  to 
resistance  being  at  the  same  time  reasonably  small.  The  highest 
e.  m.  f  that  can  be  satisfactorily  used  upon  a  pole  line  exposed 
to  the  air  is  about  40,000  volts,  inasmuch  as  the  leakage  from 
wire  to  wire  (outgoing  and  returning  wires)  in  the  form  of  brush 
or  spark  discharge  becomes  excessive  at  about  60,000  volts  un- 
less the  wires  are  very  large  and  very  far  apart.  For  transmis- 
sion within  a  radius  of  two  or  three  miles  1000  and  2000  volts 
are  usually  employed. 

151.  Power  and  e.  m.  f.  loss  in  line. — If,  say,  10^  of  the  power 
output  of  a  direct-current  dynamo  is  lost  in  the  line,  then  lofo  of 
the  e.  m.  f.  of  the  dynamo  is  also  lost  in  the  line  and  90/0  only 

(198) 


TRANSMISSION   LINES.  1-99 

is  effective  at  the  receiving  circuit.  With  alternating  current, 
however,  the  receiving  circuit  may  receive,  say,  90^  of  the  power 
output  of  the  dynamo,  while  the  effective  e.  m.  f.  at  the  receiving 
circuit  may  be  more  or  less  than  90/0  of  the  e.  m.  f.  of  the 
dynamo.  The  difference  (numerical)  between  dynamo  e.  m.  f. 
and  the  e.  m.  f.  at  the  receiver  circuit  is  called  the  line  drop  and 
this  line  drop  is  of  more  practical  importance  than  the  power  lost 
in  the  line,  inasmuch  as  nearly  all  receiving  apparatus  needs  to 
be  supplied  with  current  at  approximately  constant  e.  m.  f.  This 
is  usually  provided  for  by  over-compounding  the  dynamo  so  as 
to  keep  the  receiver  e.  m.  f.  constant.  Thus,  if  the  line  is  de- 
signed to  give  10^  drop,  the  dynamo  would  be  10^  over-com- 
pounded. 

Problem  VII.,  Chapter  VIL,  involves  the  general  question  of 
line  drop  and  the  results  of  this  problem  are  applied  to  the  calcu- 
lation of  transmission  lines  to  give  a  prescribed  drop.  This  prob- 
lem may  be,  however,  considerably  simplified  for  practical  use  as 
is  shown  later. 

152.  Line  resistance. — The  resistance  of  a  wire  for  alternating 
currents  may  in  all  practical  cases  be  taken  to  be  the  same  as  the 
resistance  of  the  same  wire  for  direct  current.  The  fact  is,  how- 
ever, that  the  alternating  current  near  the  axis  of  a  wire  lags  in 
phase  behind  the  current  near  the  surface  of  the  wire,  and  the 
resistance  of  the  wire  is  therefore  larger  for  an  alternating  cur- 
rent than  for  a  direct  current. 

153.  Line  reactance. — The  reactance  of  a  transmission  line  (out- 
going and  returning  wires  side  by  side)  is  greater  the  smaller  the 
wires  and  the  further  they  are  apart,  and  is  proportional  to  the 
length  of  the  line  and  to  the  frequency.  The  accompanying  table 
gives  the  resistance  and  reactance  per  half  mile  of  transmission 
line. 


200      THE   ELEMENTS   OF   ALTERNATING   CURRENTS. 


Resistance  and  Reactance  of  One  Mile  of  Wire  {)4  Mile  of  Transmis- 
sion Line)  (Emmet). 


K 

Reactance 

IN  Ohms. 

Size  of 

esis- 

At  60  cycles  per 

jec. 

At  125  cycles  per  sec. 

tai 
o 

S  gauge. 

ims.            1 

iVires 

Wires 

Wires 

Wires 

Wires 

Wires 

12 

inches 

18  inches 

24  inches 

12  inches 

18  inches 

24  inches 

£ 

ipart. 

apart. 

apart. 

apart. 

apart. 

apart. 

GOOD 

•259 

508 

•557 

•591 

1.06 

1. 17 

1.23 

GOO 

324 

523 

573 

.607 

09 

1.20 

1.26 

GO 

412 

534 

588 

.618 

12 

1.23 

1.29 

G 

519 

550 

603 

■633 

15 

1.26 

1.32 

I 

655 

5&5 

614 

.648 

18 

1.28 

^•35 

2 

826 

580 

629 

.663 

21 

I-3I 

1.38 

3 

I 

G4I 

591 

644 

.674 

24 

1-34 

1. 41 

4 

I 

313 

bo6 

656 

.690 

26 

1-37 

1.44 

5 

I 

656 

620 

670 

.704 

30 

1.40 

1.47 

6 

2 

088 

^33 

685 

.720 

32 

1-43 

1.49 

7 

2 

633 

647 

700 

•730 

35 

I  46 

1.52 

8 

3 

320 

662 

712 

.742 

38 

1.48 

1-55 

9 

4 

186 

677 

727 

.761 

141 

I-5I 

1.58 

lO 

5 

280 

688 

742 

.776 

1.44 

1-54 

1.62 

154.  Line  capacity. — The  two  wires  of  a  transmission  line  con- 
.stitute  a  condenser,  and  they  are  repeatedly  charged  and  dis- 
charged with  the  pulsations  of  e.  m.  f  The  current  (charging 
current)  which  the  lines  take  from  the  generator  to  charge  and 
discharge  them  is  approximately  90°  ahead  of  the  e.  m.  f.  in  phase, 
and  if  the  major  part  of  the  line  current  ia  behind  the  e.  m.  f.  in 
phase,  as  it  usually  is,  the  effect  of  the  charging  current  is  to 
slightly  lessen  the  total  current  taken  from  the  generator.  This 
charging  current  is  usually  ignored  in  the  calculation  of  transmis- 
sion lines.  The  student  will  find  a  discussion  of  its  effects  in 
Chapter  XII.  of  Steinmetz's   "  Alternating  current  phenomena." 

155.  Interference  of  separate  transmission  lines. — When  more 
than  one  transmission  line  (more  than  two  wires)  is  strung  on  the 
same  poles  the  alternating  current  in  each  line  induces  e.  m.  f 's 
in  the  other  lines  and  affects  the  line  drop.  This  interference  of 
one  line  upon  another  is  obviated  by  crossing  the  lines  at  every 
second  or  third  pole  as  shown  in  Figs.  183,  184  and  185.  Fig. 
183  shows  the  arrangement  of  a  single-phase  alternating  current 
Hne  to  avoid  inductive  effects  upon  any  other  line  that  may  be  in 


TRANSMISSION   LINES.  20I 

the  neighborhood  ;  Fig.  1 84  shows  the  arrangement  of  four  wires 
for  transmitting  two-phase  currents  ;  and  Fig.  1 8  5  shows  the  ar- 
rangement of  three  wires  for  transmitting  three-phase  currents. 


X  X  X 

Fig.    183. 

Remark :  Transmission  lines  also  affect  neighboring  lines  by 
charging  and  discharging  them  electrostatically  with  the  pulsations 
of  e.  m.  {.;  and  by  leakage  currents  due  to  incomplete  insulation. 

156.  Calculations  of  a  transmission  line  to  give  a  specified  line 
drop  {single-pJiase). — A  transmission  line  is  usually  designed  to 
deliver  a  prescribed  amount  of  power  P  at  prescribed  e.  m.  f,  E 


X        X        X- 


yc^zyzuzyQ 


Fig.  184. 

to  a  receiver  circuit  of  which  the  power  factor,  cos  d  [see  Art. 
48] ,  is  given.  The  line  drop,  frequency,  length  of  line  and  dis- 
tance apart  of  wires  are  also  given. 

The  generator  e.  m.  i.  E^  is  equal  to  the  sum  (numerical  sum) 
of  E  and  line  drop. 

The  full  load  current  /  is  found  from  EI  cos  d  =  P. 


Fig.  185. 

The  component  of  E  parallel  to  /  is  ^cos  d,  and  the  compo- 
nent of  E  perpendicular  to  I  \a  E  sin  d. 

By  treating  the  problem  at  first  as  a  direct  current  problem  the 
approximate  resistance  r'  of  the  line  is  found,  namely,  r'/=  line 
drop.     From  this  approximate  resistance  and  length  of  line  the 


202      THE   ELEMENTS   OF  ALTERNATING   CURRENTS. 

approximate  size  of  wire  and  line  reactance  x  is  found  from  the 
table ;  and  since  the  line  reactance  varies  but  little  with  size  of 
wire  the  value  of  x  need  not  be  further  approximated. 

The  component  of  E^  parallel  to  I  is  E  cos  d  -\-  rl  where  r  is 
the  true  resistance  of  the  line,  and  the  component  of  E^  perpen- 
dicular to  I  is  E  sin  d  +  xl.     Therefore 

E^^  =  (E  cos  d  +  rif  +  {Esin  d  +  xlf 

or  r= 5 ^ y '- (114) 

From  this  equation  the  true  line  resistance  r  may  be  found  and 
thence  the  correct  size  of  wire. 
Example  : 

E=  20,000  volts, 

P=  1,000  kilowatts, 

Cos  ^  =  .85  =  power  factor  of  receiving  circuit, 

j5"p  =  23,000  volts  or  line  drop  =  3,000  volts, 

frequency  =  60  cycles  per  second, 

distance  =  30  miles, 

distance  apart  of  wires  =18  inches. 
From  these  data  we  find  : 

/=  58.8  amperes. 

r'  =  51  ohms. 
Therefore,  from  the  table  we  find  that,  approximately,  a  No. 
2  B.  &  S.  wire  is  required  so  that  x=^  37.7  ohms. 
Further 

Ecos  d  =  17,000  volts, 

^sin  ^  +  xl==  12,700  volts, 
and  from  equation  (114)  we  find 

r=  37.3  ohms 
from  which  the  correct  size  of  wire  is  found  to  be  approximately 
a  No.  I  B.  &  S. 

157.  Calculation  of  double  line  for  two-phase  transmission  (four 
<wires). — In  this  case  each  line  is  calculated  to  deliver  half  the 


TRANSMISSION   LINES. 


203 


Receiver 


Fig.  186. 


prescribed  power.  Thus,  if  it  is  desired  to  deliver  1,000  kilowatts 
at  20,000  volts  two-phase,  at  a  frequency  of  60,  line  drop  of 
3,000  volts,  etc.,  then  each  line  is  calculated  as  a  single-phase 
line  to  deliver  500  kilowatts  at  3,000  volts  line  drop.  The  Hnes 
being,  of  course,  arranged  as  shown  in  Fig.  184. 

158.  Calculation  of  a  three-wire  transmission  line  for  three-phase 
currents. — The  calculation  will  be  carried  out  for  the  case  of  Y- 
connected  generator  and 

y^connected  receiver  as      y- r 

shown  in  Fig.  186,  for 
the  reason  that  the  rela- 
tion between  ^Sj,,  E,  and 
line  current  is  then  the 
simplest. 

Let  cos  d  be  the  power  factor  of  each  receiving  circuit,  P  the 
total  power  to  be  delivered,  E  the  e.  m,  f  between  the  terminals 
of  each  receiving  circuit  and  E^  the  e.  m.  f.  of  each  armature 
winding  on  the  generator*;  all  prescribed. 

Then  P=  ^EIcos  6 

from  which  the  full  load  line  current  /  may  be  calculated. 

The  difference  ^^  —  ^  is  due  to  e.  m.  f.  drop  in  one  main. 
Therefore,  looking  upon  the  problem  as  one  in  direct  currents,  we 
have  E^  —  E  =  r'l  where  r'  is  the  approximate  resistance  of  one 
main.  From  this  the  approximate  size  of  wire  may  be  found 
from  the  table. 

Consider  one  of  the  mains ;  say,  main  No.  2  ;  the  other  two 
mains  together  constitute  the  return  circuit  for  this  main,  and  the 
average  distance  from  main  2  to  mains  i  and  3  is  i|^/  when  the 
mains  are  crossed  as  shown  in  Fig.  185.  Find  the  reactance  x 
of  a  pair  of  mains  each  of  the  size  approximated  above  and  dis- 
tant i^l  from  each  other. 

*  The  e.  m.  f.  between  mains  at  receiving  station  is  1/3  E  and  the  e.  m.  f.  between 
mains  at  generating  station  \s  V  2,  -Eq. 


204   THE  ELEMENTS  OF  ALTERNATING  CURRENTS. 

The  component  of  E  parallel  to  /  is  Ecosd  and  the  com- 
ponent of  E  perpendicular  to  I  is  E  sin  d. 

The  resistance  drop  in  one  main  is  rl  and  the  reactance  drop 
in  one  main  is  ixl,  the  former  being  parallel  to  /  and  the  latter 
perpendicular  to  /. 

Then  the  components  of  E^  are  Ecos  d  +  rl  and  ^sin  ^  +  i^I 
so  that 

E^^  =  (Ecos  d  +  rIY  +  (^sin  d  +  lxl)\ 


or  r= 9 V -^ i ^  (115) 

which  gives  r,  the  true  resistance  of  one  main,  from  which  the 
correct  size  of  wire  is  easily  found. 

The  calculation  of  a  transmission  line  when  the  e.  m.  f.'s  be- 
tween mains  is  specified  instead  of  the  e.  m.  f.'s  in  F-connected 
circuits  is  sufficiently  explained  in  the  following  example. 

Example:  E.  m.  f  between  mains  at  receiving  station  to  be 
20,000  volts.  Therefore  e.  m.  f  between  terminals  of  F-con- 
nected receiving  circuits  would  be  20,000  -r-  v^S.     Therefore 

E=  11,550  volts. 

E.  m.  i.  between  mains  at  generating  station  to  be  23,000  volts. 
Therefore  E^  =  23,000  -^  v^s  or  : 

Eq  =  13,280  volts. 

Further  specifications  : 
P=  1,000  kilowatts, 
Cos/?  =  .85, 

frequency  =  60  cycles  per  second, 
distance  =  30  miles, 

distance  apart  of  adjacent  wires  =15^  inches  (  =  /). 
From  these  data  we  find 

/=  34.0  amperes, 
/  =  50.9  ohms. 


TRANSMISSION   LINES.  20$ 

Therefore,  approximately,  a  No.  5  wire  is  needed.  The  react- 
ance, X,  of  a  30-mile  double  line  of  No.  5  wires  at  21  inches 
(=  i^/)  apart  is,  from  the  table, 

X  =  41.2  ohms. 
Equation  (115)  then  gives 

r  =  46.5  ohms. 

So  that  a  wire  between  No.  4  and  No.  5  would  give  the  pre- 
scribed line  drop. 

THE  END. 


INDEX. 


Absolute  electrometer,  the,  33. 
Admittance,  definition  of,  67. 
Air  gap,  magnetic  densities  in,  113. 
All-day  efficiency  of  the  transformer,  139. 
Alternating  currents,  advantages  of,  80. 
Alternating    current,    utilization   of,  foi 

motive  purposes,  178. 
Alternator,  armature  of,   19. 
Alternator,  brushes  of,  19. 
Alternator,  characteristic  curve  of,  98. 
Alternator,  collecting  rings  of,   1 9. 
Alternator,  the  constant  current,  98. 
Alternator,  exciter  of,  19. 
Alternator,  field  magnet  of,  18. 
Alternator,  fundamental  equation  of,   24. 
Alternator,  number  of  poles,  104. 
Alternator  output,  limits  of,  loi. 
Alternator,  phase  constant  of,   100. 
Alternator,   polyphase,    as  an  induction 

motor,   197. 
Alternator,  the  simple,  18. 
Alternator,  speed  and  frequency  of,  19. 
Alternator  speeds,  104. 
Alternator,  the  single-phase,  80. 
Alternator,  the  two-phase,  81. 
Alternator,  the  three-phase,  84. 
Alternators,  compounding  of,   1 16. 
Alternators,  design  of,  114. 
Alternators,  field  excitation  of,   115. 
Alternators,  in  parallel,  160. 
Alternators  in  series,  151,  160. 
Ammeter,  the  electrodynamometer,  29. 
Ammeter,  the  hot  wire,  29. 
Ammeter,  the  plungei  type,  33. 
Analogies,  mechanical  and  electrical,  15. 
Armature  conductor,  current  densities  in, 

US- 
Armature  core,  magnetic  densitijes  in,  1 13. 


Armature  current,  magnetizing  action  of, 

95- 
Armature  current,  reaction  of,  95. 
Armature  drop,  97. 
Armature  inductance,  96. 
Armature  reaction,  95. 
Armature  of  simple  alternator,  19. 
Armature  windings,  105. 
Armature  windings,  single-phase,  106. 
Armature  windings,  two-phase,  107. 
Armature  windings,  three-phase,  108. 
Armatures,  disk,  104. 
Armatures,  drum,  104. 
Armatures,  insulation  of,  112. 
Armatures,  ring,  105. 
Average  values  of  e.  m.  f.  and  current, 

23- 
Average  value  of  harmonic  e.  m.  f.  and 

current,  47. 
Average  values,  definition  of,  46. 

Bar  winding.  III. 

Cardew's  voltmeter,  29. 

Characteristic  curve  of  alternator,  98. 

Collecting  rings  of  alternator,  1 9. 

Complex  quantity,  addition  and  subtrac- 
tion of,  63. 

Complex  quantity,  definition  of,  62. 

Complex  quantity,  multiplication  and  di- 
vision of,  63. 

Complex  quantity,  the  use  of,  62. 

Composition  and  resolution  of  harmonic 
e.  m.  f.'s  and  currents,  42. 

Compounding  of  alternators,  1 16. 

Condenser,  the,  14. 

Conductance,  definition  of,  67. 

Constant  current  alternator,  98. 


(207) 


208 


INDEX. 


Constant  current  transformer,  1 35. 
Contact  maker,  the,  26. 
Converter,  the  rotary,  166. 
Converter,  armature  current  of,  170, 
Converter,  armature  reaction  of,  170. 
Converter,  armature  heating  of,  175. 
Converter,  current  relations  of,  174. 
Converter,  e.  m.  f,  relations  of,  1 72. 
Converter,  hunting  of,  or  pumping  of,  169. 
Converter,  operation  of,  168. 
Converter,  rating  of,  171. 
Converter,  starting  of,  168. 
Converter,  single-phase,  i65. 
Converter,  two  phase,  166. 
Converter,  three-phase,  166. 
Converter,  use  of,  167. 
Copper  losses  of  transformer.  138. 
Core  flux,  maximum  value  of,  123. 
Core  reluctance,  effect   of,    on  action  ot 

transformer,  124. 
Coulomb,  definition  of,   13. 
Current  and  e.  m.  f.  curves,  20. 
Current  densities,  113. 
Curves  of  e.  m.  f.  and  current,  20. 
Cycle,  definition  of,  41. 

Decay  and  grov^th  of  current,  9. 
Decaying  oscillatory  circuit,  56. 
Definition  of  admittance,  67. 
Definition  of  average  or  mean  value,  46. 
Definition  of  complex  quantity,  62. 
Definition  of  conductance,  67. 
Definition  of  the  coulomb,  13. 
Definition  of  cycle,  41. 
Definition  of  electric  charge,  13. 
Definition  of  electrostatic  capacity,  14. 
Definition  of  the  farad,  14. 
Definition  of  frequency,  41. 
Definition  of  harmonic  current,  39. 
Definition  of  harmonic  e.  m.  f.,  39. 
Definition  of  the  henry,  4. 
Definition  of  impedance,  66. 
Definition  of  inductance,  3. 
Definition  of  magnetic  flux,    I. 
Definition  of  the  microfarad,  I4. 
Definition  of  mutual  inductance,  12. 


Definition  of  opposition,  41. 
Definition  of  period,  41. 
Definition  of  phase  constant,  loo. 
Definition  of  phase  difference,  4I. 
Definition  of  pow^er  factor,  50. 
Definition  of  quadrature,  41. 
Definition  of  reactance,  66. 
Definition  of  resistance,  66. 
Definition  of  simple  quantity,  62. 
Definition  of  shp  of  induction  motor,  189. 
Definition  of  susceptance,  67. 
Definition  of  synchronism,  41. 
Definition  of  vector  quantity,  63. 
Delta  connection  for  three-phase  systems, 

87,  91. 
Design  of  alternators,  1 14. 
Design  of  induction  motor,  196. 
Design  of  transformers,  140. 
Design  of  transmission  lines,  201. 
Distortion  of  field  by  armature  current, 

95- 
Distributed  winding,  effect  of,  99. 

Eddy  current  loss  in  iron  cores,  137. 

Eddy  currents,  effect  of,  on  action  of  a 
transformer,  1 24. 

Effective  values  of  e.  m.  f.'s  and  cur- 
rents, 23. 

Effective  values  of  harmonic  e.  m.  f.'s 
and  currents,  48. 

Efficiency  of  alternators,  140. 

Efficiency  of  induction  motor,  185. 

Efficiency   of    synchronous   motor,    I55> 

157- 
Efficiency  of  transformers,  138. 
Electrical  and  mechanical  analogies,  15. 
Electric  charge,  definition  of,  13. 
Electric  charge,  measurement  of,  13. 
Electric  resonance,  57. 
Electrodynamometer,  the,  29. 
Electrometer,  the  absolute,  33. 
Electromotive  force  and  current  curves, 

20. 
Electromotive  force  drop  in  transmission 

lines,  198. 
Electromotive  force,  induced,  2. 


INDEX. 


209 


Electromotive  force,  lost  in  armature,  97. 
Electromotive  force,  self-induced,  7. 
Electrostatic  capacity,  definition  of,  14. 
Electrostatic  voltmeter,  the,  32. 
Equation,  fundamental,  of  tlie  alternator, 

24. 
Equivalent  resistance  and   reactance  ot 

transformer,  122 
Excitation  characteristics  of  synchronous 

motor,  163. 
Exciter  of  alternator,  19. 

Farad,  the,  definition  of,  14. 
Field  excitation  of  alternators,  US- 
Field  magnet  of  alternator,  18. 
Flux,  magnetic,  definition  of,  I. 
Form  factor  of  alternating  e.  m.  f.,  49. 
Frequency  and  speed,  relation  of,  19. 
Frequency  changer,  the,  189. 
Frequency,  definition  of,  41. 
Frequencies  of  alternators,  103. 

Graphical  method,  62. 

Growth  and  decay  of  current,  9. 

Harmonic  current,  definition  of,  39. 

Harmonic  currents,  composition  and 
resolution  of,  42. 

Harmonic  e.  m.  f. ,  definition  of,  39. 

Harmonic  e.  m.  f.'s,  composition  and 
resolution  of,  42. 

Harmonic  e.  m.  f.'s  and  currents,  aver- 
age values  of,  47. 

Harmonic  e.  m.  f 's  and  currents,  effec- 
tive values  of,  48. 

Harmonic  e.  m.  f.'s  and  currents,  rates 
of  change  of,  44. 

Henry,  the  definition  of,  4. 

Hot  wire  ammeter  and  voltmeter,  29. 

Hysteresis,  effect  of,  on  action  of  a  trans- 
former,  124. 

Hysteresis  loss  in  iron  cores,  137. 

Impedance,  definition  of,  66. 
Induced  electromotive  force,  2. 
Inductance,  calculation  of,  8. 


Inductance,  definition  of,  3. 

Inductance,  mechanical  analogue  of,  5. 

Inductance  of  armature,  96. 

Induction  generator,  186. 

Induction  motor,  the,  178. 

Induction  motor,  action  of,  as  a  transfor- 
mer, 188. 

Induction  motor,  calculation  of  leakage 
inductance  of,  196. 

Induction  motor,  effect  of  eddy  currents 
and  hysteresis,  191. 

Induction  motor,  effect  of  magnetic  leak- 
age, 192. 

Induction  motor,  effect  of  speed  upon 
transformer  action  of,  188. 

Induction  motor,  effect  of  stator  resist- 
ance, 192. 

Induction  motor,  efficiency  and  rotor  re- 
sistance, 185. 

Induction  motor,  efficiency  and  speed, 
185. 

Induction  motor,  general  theory  of,  137. 

Induction  motor,  maximum  torque  of,  195. 

Induction  motor,  maximum  starting  tor- 
que, 196. 

Induction  motor,  polyphase  alternator  as 
an,  197. 

Induction  motor,  single-phase  driving, 
186. 

Induction  motor,  slip  of,  189. 

Induction  motor,  starting  torque  of,  195. 

Induction  motor,  stator  windings  of,  179. 

Induction  motor,  the  actual,  I91. 

Induction  motor,  the  ideal,  190. 

Induction  motor,  torque  and  speed,  184. 

Induction  motor  used  as  a  frequency 
changer,  189. 

Induction  motor,  use  of  starting  resist- 
ance, 184. 

Induction  motors,  designing  of,  196. 

Induction  motors,  use  of  two,  for  street 
railway  work,    187. 

Insulation  of  armatures,  112. 

Iron  losses  of  transformer,  137. 

Kinetic  energy  of  the  electric  current    3. 


2IO 


INDEX. 


Leakage  current  of  transformer,  125. 
Leakage  inductance  of  a  transformer,  131. 
Leakage,  magnetic,  effect  of,  upon  trans- 
former, 139. 
Line  drop,  198. 

Magnetizing  current  of  transformer,  125. 
Magnetic  densities  in  armature  and  air 

gap,  113- 

Magnetic  densities  for  transformer  cores, 
141. 

Magnetic  flux,  definition  of,  I. 

Magnetic  flux,  through  a  coil,  2. 

Magnetic  leakage,  eff^ect  of,  upon  trans- 
former, 130. 

Mean  values,  definition  of,  46. 

Measurement  of  electric  charge,  13. 

Measurement  of  power,  35. 

Mechanical  and  electrical  analogies,  15. 

Mechanical  resonance,  59. 

Mesh  connection  for  three-phase  system, 
87.  91. 

Microfarad,  the,  definition  of,  14. 

Moment  of  inertia,  analogue  of  induct- 
ance, 5- 

Monocyclic  system,  148. 

Motor,  induction.     See  induction  motor. 

Motor,  the  synchronous.  See  synchro- 
nous motor. 

Mutual  inductance,  definition  of,  12. 

Noninductive  circuits,  4. 

Opposition,  definition  of,  41. 
Oscillatory  current,  the,  56. 
Oscillatory  current,  the  decaying,  56. 
Output,  influence  of  inductance  upon,  103. 
Output,  limits  of,  of  alternator,  loi. 
Output,  limit  of  transformer,  139. 

Period,  definition  of,  41. 
Phase  constant,  definition  of,  loo. 
Phase  constants,  table  of,  loi. 
Phase  difference,  definition  of,  41. 
Plunger    type,   ammeter  and  voltmeter, 
33- 


Primary  of  transformer,  1 1 9. 

Problem  I,  9. 

Problem  II,  9. 

Problem  III,  51. 

Problem  IV,  51,  66. 

Problem  V,  53. 

Problem  VI,  54,  66. 

Problem  VII,  69. 

Problem  VIII,  73. 

Problem  IX,  75. 

Problem  X,  76. 

Poles  of  alternators,  number  of,  104. 

Power  developed  by  harmonic  e.  m.  f..  49. 

Power  factor,  definition  of,  50. 

Power,  instantaneous  and  average,  values 

of,  22. 
Power  in  polyphase  systems,  91. 
Power,  measurement  of,  35. 
Power      measurement,     three  -  ammeter 

method,  36. 
Power      measurement,      three-voltmeter 

method,  35.  " 

Quadrature,  definition  of,  41. 

Rates  of  change  of  harmonic  e.  m.  f.'s 
and  currents,  44. 

Rating  of  transformers,  140. 

Ratio  of  transformation  of  transformer, 
119. 

Reactance,  definition  of,  66. 

Reaction  of  armature  currents,  95. 

Recording  wattmeter,  the,  38. 

Rectifying  commutator,  1 16. 

Regulation  of  the  transformer,  127. 

Regulation  of  the  transformer,  calcula- 
tion of,  134. 

Resistance,  definition  of,  66. 

Resolution  and  composition  of  harmonic 
e.  m.  f.'s  and  currents,  42. 

Resonance,  electric,  57. 

Resonance,  mechanical,  59- 

Rotary  converter,  the.     See  converter. 

Rotor,  the  squirrel  cage,  179. 

Rotor,  definition  of,  179. 


INDEX. 


211 


Scott's  transformer,  148. 

Secondary  of  transformer,  II9. 

Self-induced  electromotive  force,  7. 

Simple  quantity,  definition  of,  62. 

Single-phase  armature  windings,  106. 

Slip  of  induction  motor,  189. 

Spark  gauge,  the,  33. 

Speed  and  frequency,  relation  of,  19. 

Speeds  of  alternators,  104. 

Squirrel  cage  rotor,  179. 

Stator,  definition  of,  178. 

Stator  windings  of  induction  motor,  179. 

Steinmetz'  method,  62. 
■  Steinmetz  system  of  tandem  control  for 
induction  motors,  1 87. 

Susceptance,  definition  of,  67. 

Symbolic  method,  application  of,  66. 

Synchronism,  definition  of,  41. 

Synchronous  motor,  the,  151. 

Synchronous  motor,  excitation  charac- 
teristic of,  163. 

Synchronous  motor,  fundamental  equa- 
tions of,  154. 

Synchronous  motor,  greatest  e.  m.  f.  of, 
161. 

Synchronous  motor,  greatest  intake  of, 
161,  162. 

Synchronous  motor,  hunting  of,  or  pump- 
ing of,  169. 

Synchronous  motor,  stability  of,  159. 

Synchronous  motor,  starting  of,  157. 

Table  of  line  resistance  and  reactance, 
200. 

Table  magnetic  densities  for  transformer 
cores,  141. 

Table  of  phase  constants,  loi. 

Table  of  transformer  efficiencies,  138. 

Thomson  inclined  coil  ammeter  and  volt- 
meter, 34. 

Thomson  recording  wattmeter,  38. 

Three- ammeter  method,  36. 

Three-phase  armature  windings,  108. 

Three-phase  alternator,  84. 

Three-phase  e.  m.  f.'s  and  currents,  83, 
85- 


Three-phase  system,  balanced,  90. 
Three-phase  system,  e.  m.  f.  and  current 

relations,  88. 
Three  phase  system,  power,  91,  93. 
Three-voltmeter  method,  35. 
Transformation,  ratio  of,  of  transformer, 

119. 
Transformer,  the  actual,  124. 
Transformer,  the  ideal,  119. 
Transformer  connections,  I43. 
Transformer,  constant  current,  135. 
Transformer  cores,  magnetic  densities  for, 

141. 
Transformer  design,  I40. 
Transformer,  effect  of  coil  resistances,  1 29. 
Transformer,  effect  of  core  reluctance  and 

hysteresis  upon,  1 24. 
Transformer,  effect  of  magnedc  leakage, 

ISO- 
Transformer  efficiency,  138. 
Transformer  efficiency,  all  day,  139. 
Transformer  losses,  1 37. 
Transformer,     magnetizing    or    leakage 

current  of,  1 25. 
Transformer  output,  limits  of,  139. 
Transformer,  primary  and  secondary  of, 

119. 
Transformer  rating,  140. 
Transformer,  ratio  of  transformation,  1 1 9. 
Transformer  regulation,  127. 
Transformer    regulation,    calculation    of, 

134. 

Transformer,  Scott's,  148. 

Transformer,  the  general  alternating  cur- 
rent,  187. 

Transformer,  two-phase, three-phase,  146 

Transformers  with  divided  coils,  144. 

Transmission  lines,  198. 

Transmission  line  capacity,  200. 

Transmission  line,  e.  m.  f.  drop,  198. 

Transmission  line  reactance,  199. 

Transmission  line  resistance,  199. 

Transmission  lines,  designing  of,  201. 

Transmission  lines,  interference  of,  200. 

Trigonometrical  method,  62. 

Two-phase  armature  windings,  107. 


212 


INDEX. 


Two-phase  alternator,  8 1. 

Two-phase  system   e.  m.  f.  and  current 

relations,  84. 
Two-phase  system  balanced,  84. 
Two-phase  system,  power,  91. 

Units  of  electric  charge,  13. 
Units  of  inductance,  4. 

Vector,  addition  and  subtraction,  64. 
Vector  division,  65 
Vector  multiplication,  64. 
Vector,  numerical  value  of,  64. 
Vector  quantity,  definition  of,.  63. 


Voltmeter,  the  absolute  electrostatic,  33. 
Voltmeter,  Cardew's,  29. 
Voltmeter,  the  electrodynamometer,  31. 
Voltmeter,  the  electrostatic,  32. 
Voltmeter,  the  hot  wire,  29. 
Voltmeter,  the  plunger  type,  33. 
Voltmeter,  the  spark  gauge,  33. 

Wattmeter,  the,  36. 
Wattmeter,  the  recording,  38. 
Windings  of  armatures,  105. 

Y-connection  for  three-phase  system,  87, 
90. 


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